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# Least-action classical electrodynamics without potentials

Is it possible to formulate classical electrodynamics (in the sense of deriving Maxwell's equations) from a least-action principle, without the use of potentials? That is, is there a lagrangian which depends only on the electric and magnetic fields and which will have Maxwell's equations as its Euler-Lagrange equations?

• I really can't see the connection between the derivation of Maxwell Eq. from the least action principle and to not use potentials, we can include or remove potentials at any point, because in classical physics (not quantum) this is just a matter of definitions. Regarding the least action derivation, to use it we need to have some Lagrangian to start with, if you suppose it given then off course you can derive all Maxwell Equations. Anyway an interesting thing I know, that if you will just suppose that we have some force that behaves like gravity (central and proportional to the distances squar
– TMS
Feb 4, 2013 at 17:33
• Going through the answers, Occam's razor tells me this is not the way to go. Feb 28, 2021 at 10:47
• Apr 2, 2021 at 13:49

1) Well, at the classical level, if we are allowed to introduce auxiliary variables, we can always trivially encode a set of equations of motion $$\tag{1} {\rm eom}_i = 0, \qquad i\in\{1, \ldots, n\},$$ in a brute force manner with the help of Lagrange multipliers $$\tag{2}\lambda^i, \qquad i\in\{1, \ldots, n\},$$ so that the Lagrangian density simply reads $$\tag{3}{\cal L}~=~\sum_{i=1}^n\lambda^i ~{\rm eom}_i.$$

This is for many reasons not a very satisfactory solution. (Especially if we start to think about quantum mechanical aspects. However, OP only asks about classical physics.) Nevertheless, the above trivial rewritings (3) illustrates how it is hard to formulate and prove no-go theorems with air-tight arguments.

2) To proceed, we must impose additional conditions on the form of the action principle. Firstly, since we are forbidden to introduce gauge potentials $A_{\mu}$ as fundamental variables (that we can vary in the action principle), we will assume that the fundamental EM variables in vacuum should be given by the ${\bf E}$ and ${\bf B}$ field. Already in pure EM, it is impossible to get the $1+1+3+3=8$ Maxwell eqs. (in differential form) as Euler-Lagrange eqs. by varying only the $3+3=6$ field variables ${\bf E}$ and ${\bf B}$. So we would eventually have to introduce additional field variables, one way or another.

3a) It doesn't get any better if we try to couple EM to matter. In decoupling corners of the theory, we should be able to recover well-known special cases. E.g. in the case of EM coupled to charged point particles, say in a non-relativistic limit where there is no EM field, the Lagrangian of a single point charge should reduce to the well-known form

$$\tag{4}L~=~\frac{1}{2}mv^2$$

of a free particle. A discussion of eq. (4) can be found e.g. in this Phys.SE post. Here we will assume that eq. (4) is valid in what follows.

3b) Next question is what happens in electrostatics

$$\tag{5} m\dot{\bf v}~=~ q{\bf E}?$$

$$\tag{6} L~=~\frac{1}{2}mv^2 - V$$

with potential energy

$$\tag{7}V~=~ q\phi,$$

where $\phi$ is the scalar electric potential. However, since we are forbidden to introduce the potential $\phi$ as a fundamental variable, we must interpret it

$$\tag{8}\phi({\bf r})~:=~-\int^{\bf r} \!d{\bf r}^{\prime}\cdot{\bf E}({\bf r}^{\prime})$$

as a functional of the electric field ${\bf E}$, which in turn is taken as the fundamental field. Note that eqs. (6)-(8) correspond to a non-local action.

3c) The straightforward generalization (from point mechanics to field theory) of eq. (7) is a potential density

$$\tag{9}{\cal V}~=~ \rho\phi,$$

where $\rho$ is an electric charge density. Readers familiar with the usual action principle for Maxwell's eqs. will recognize that we are very close to argue that the interaction term between pure EM and matter must be of the form

$$\tag{10} {\cal L}_{\rm int}~=~J^{\mu}A_{\mu},$$

even if we haven't yet discussed what should replace the standard Lagrangian

$$\tag{11} {\cal L}_{\rm EM} ~=~-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}$$

for pure EM.

3d) Staying in electrostatics, let us ponder our prospects of deriving Gauss' law in differential form

$$\tag{12} \nabla \cdot {\bf E} ~=~ \rho.$$

Obviously, the rhs. of the single eq. (12) should appear by varying the potential density (9) wrt. one of the three $E$ fields, but which one? The counting is not right. And because eq. (9) is non-local, we will in any case get an integrated version of $\rho$ rather than $\rho$ itself, which appears on the rhs. of eq. (12), and which we set out to reproduce.

3e) In conclusion, it seems hopeless to couple a EM theory (with ${\bf E}$ and ${\bf B}$ as fundamental variables) to matter, and reproduce standard classical eqs. of motion.

4) The standard remedy is to introduce $4$ (globally defined) gauge potentials $A_{\mu}$ as fundamental variables. This makes $1+3=4$ source-less Maxwell eqs. trivial, and the remaining $1+3=4$ Maxwell eqs. with sources may be derived by varying wrt. the $4$ fundamental variables $A_{\mu}$.

For instance, the standard (special relativistic) action for EM coupled to $n$ massive point charges $q_1, \ldots, q_n$, at positions ${\bf r}_1, \ldots, {\bf r}_n$, is given as

$$\tag{13} S[A^{\mu}; {\bf r}_i] ~=~\int \! dt ~L,$$

where the Lagrangian is

$$\tag{14} L ~=~ -\frac{1}{4}\int \! d^3r ~F_{\mu\nu}F^{\mu\nu} -\sum_{i=1}^n \left(\frac{m_{0i}c^2}{\gamma({\bf v}_i)} +q_i\{\phi({\bf r}_i) - {\bf v}_i\cdot {\bf A}({\bf r}_i)\} \right).$$

The corresponding Euler-Lagrange eqs. are $4$ Maxwell eqs. with sources (when varying $A_{\mu})$, and $n$ (special relativistic) Newton's 2nd laws with Lorentz forces (when varying ${\bf r}_i)$.

• Thank you @Qmechanic! Does it mean that even given the equations of motion (such as Maxwell equations for EM field dynamics and Lorentz force equations for charge dynamics) we cannot reproduce the Lagrangian from equations of motion? For instance, if we allow for non-zero magnetic charges, then we cannot use potentials, but we still have both Maxwell and Lorentz force equations valid (of course, adjusted to existence of magnetic charges). Can we formulate least action principle in this case? Feb 5, 2013 at 6:49
• I updated the answer. The question concerning action principles for magnetic monopoles is a huge separate topic, and should be posted as a separate question. Besides OP's other Phys.SE question, a related topic of potentials in presence of magnetic monopoles has also been asked in this Phys.SE post and their links. Feb 5, 2013 at 14:48

I don't know if other approach is possible but this one does not workd, We start with tensor $F_{\mu\nu}$:

$$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$$

but forget about the 4-potential and define it to be:

$$F^{\mu\nu} = \begin{bmatrix} 0 &-E_x/c & -E_y/c & -E_z/c \\ E_x/c & 0 & -B_z & B_y \\ E_y/c & B_z & 0 & -B_x \\ E_z/c & -B_y & B_x & 0 \end{bmatrix}$$

and write the Lagrangian density as a function of the Cartesian components of the fields, say:

$$\mathcal{L}=\mathcal{L}(E_{x},..,B_{z})$$

and

$$\mathcal{L}=-\frac{1}{\mu_0}F^{\mu\nu}F_{\mu\nu}$$

Then Euler Lagrange equations gives you (for example applied to $E_{x}$) $\displaystyle \frac{\partial \mathcal{L}}{\partial{E_{x}}}=0$ so this is not consistent.

How we can solve the problem? Thinking in another Lagrangian density, defining a new $F_{\mu\nu}$ tensor, choosing more carefully the independent fields?

• The EM lagrangian is known to contain additional term $A^\mu J_\nu$. Feb 4, 2013 at 18:12

Why not try? for example with a Lagrangian of the specific form "polynomial of second order" $$\mathcal{L}(x^{\mu},X,\partial_{\mu}X):= \sum_{i,j=1}^6\left( A_{ij} X^i X ^j + \sum_{\mu=0}^3 B^{\mu}_{ij} \partial_{\mu}X^i X ^j +\sum_{\mu,\nu=0}^3 C^{\mu\nu}_{ij} \partial_{\mu} X^i \partial_{\nu} X^j \right) \tag{1}\label{1}$$ in $$X=\big(E_x, E_y,E_z, B_x, B_y, B_z \big)$$. (Since the classical fields $$X^i X^j= X^j X^i$$ commute, only the symmetric part of $$A$$ and $$C$$ play a role, so we can assume they are symmetric.) Taking into account the answer of Qmechanics, we'll not try to recover all of the Maxwell equations, but let us for example try the ones with possible "source terms": \left\lbrace \begin{aligned} \vec{\nabla}\cdot \vec{\mathbf{E}} &= 0 \\ \vec{\nabla}\wedge \frac{\vec{\mathbf{B}}}{\mu} &= \epsilon_0 \frac{\partial \vec{\mathbf{E}}}{\partial t} \end{aligned} \right. \tag{2}\label{2} With the notations of (\ref{1}), these corresponds to the vanishing of 4 linear forms (1 for the first "scalar" equation, 3 for the second vector equation; $$X^i$$ i-th component, not something to the power i ) \left\lbrace \begin{aligned} \alpha(\partial_{\mu}X)&:= \sum_{i=1}^3\sum_{\mu=0}^3 \alpha^{\mu}_{i} \partial_{\mu}X^i = \sum_{i=1}^3\sum_{\mu=0}^3 \delta^{\mu}_{i} \partial_{\mu}X^i = \sum_{i=1}^3 \partial_i X^i \\ \beta(\partial_{\mu}X)&:= \sum_{i=1}^6\sum_{\mu=0}^3 \beta^{\mu}_{i} \partial_{\mu}X^i = \frac{1}{\mu} \left(\partial_y B_z - \partial_z B_y \right) - \epsilon_0 \frac{\partial E_x}{\partial t} = \frac{1}{\mu} \left(\partial_y X^6 - \partial_z X^5 \right) - \epsilon_0 \frac{\partial X^1}{\partial t}\\ \vdots\quad &= \quad \vdots \end{aligned} \right. \tag{3}\label{3} Left-hand side of the Euler-Lagrange equation in the form $$\frac{\partial \mathcal{L}}{\partial X^i} = \partial_{\mu}\frac{\partial \mathcal{L}}{\partial (\partial_{\mu}X^i)}$$ reads $$\frac{\partial \mathcal{L}}{\partial X^i} = \sum_{j=1}^6\left(2 A_{ij} X ^j + \sum_{\mu=0}^3 B^{\mu}_{ji} \partial_{\mu}X^j \right)$$ and r.h.s. (with Einstein's summation convention) $$\partial_{\mu}\frac{\partial \mathcal{L}}{\partial (\partial_{\mu}X^i)} = \partial_{\mu} \left[\sum_{j=1}^6\left( \sum_{\mu=0}^3 B^{\mu}_{ij} X ^j + 2 \sum_{\mu,\nu=0}^3 C^{\mu\nu}_{ij} \partial_{\nu} X^j \right) \right] = \sum_{j=1}^6\left( \sum_{\mu=0}^3 B^{\mu}_{ij} \partial_{\mu}X^j + 2 \sum_{\mu,\nu=0}^3 C^{\mu\nu}_{ij} \partial_{\mu\nu} X^j \right)$$ One sees that in order to get (\ref{3}), the Lagrangian must not have terms in $$A_{ij}$$ and $$C^{\mu\nu}_{ij}$$. Then $$\frac{\partial \mathcal{L}}{\partial X^i} - \partial_{\mu}\frac{\partial \mathcal{L}}{\partial (\partial_{\mu}X^i)} = \sum_{j=1}^6 \sum_{\mu=0}^3\left( B^{\mu}_{ji} - B^{\mu}_{ij} \right)\partial_{\mu}X^j$$ (Caution: $$B^{\mu}_{ij}$$ notation from (\ref{1}), not the magnetic field...). Only its antisymmetric part plays a role so let us choose it antisymmetric to have unicity. By hand, let us set for $$i=4,\ 0 \leq \mu \leq 3$$ $$B^{\mu}_{4j}= - B^{\mu}_{j4}= \delta_j^{\mu}\enspace \text{if}\ 1\leq j \leq 3\quad \text{and}\quad B^{\mu}_{4j}= - B^{\mu}_{j4} = 0\enspace \text{if}\ 4\leq j \leq 6 \tag{4}\label{4}$$ (i.e. very explicitly the following contribution $$\ \vec{\mathbf{E}}\cdot \vec{\nabla} B_x - B_x \big(\vec{\nabla} \cdot \vec{\mathbf{E}} \big)$$ to the Lagrangian (\ref{1}))

This does yield the first equation of (\ref{3}) ($$-1$$). For $$i=2,\ 0 \leq \mu \leq 3$$, let us try $$B_{26}^2 = - B_{62}^2= \frac{-1}{2\mu}\ ,\quad B_{25}^3 = - B_{52}^3= \frac{1}{2\mu}\ ,\quad B_{21}^0 = - B_{12}^0= \frac{\epsilon_0}{2} \tag{5}\label{5}$$ and $$B_{2j}^{\mu}= B_{j2}^{\mu}=0$$ for all other $$(i=2,\mu,j)$$ different of the ones above.

(Explicit contribution to the Lagrangian: $$-\frac{1}{2\mu}\left( \partial_y E_y\, B_z - \partial_y B_z\, E_y\right) +\frac{1}{2\mu}\left( \partial_z E_y\, B_y - \partial_z B_y\, E_y\right) +\frac{\epsilon_0}{2}\left( \partial_t E_y\, E_x - \partial_t E_x\, E_y\right)$$ )

Partial conclusion: at this point there seems to be no contradiction. For each fixed $$\mu,\ (B^{\mu}_{ij})$$ is a $$6\times 6$$ antisymmetric matrix, so 5 independent coefficients and $$20$$ all in all. Imposing to recover (\ref{2}-\ref{3}) should naively reduce the possibilities to a vector space of dimension $$20-4=16$$ but the examples (\ref{4}-\ref{5}) show that one should probably think twice.

• I'm pretty sure I read in Arxiv a paper where something like this was already done. Unfortunately, I forgot the name and I'm been unable to find it with a quick search. Sep 30, 2020 at 20:46
• Ok, i'll look for it if I have time and replace this comment by a link to that article Oct 1, 2020 at 11:59

In classical electrodynamics, the physical quantities of interest are the fields. The theory is already formulated "without" potentials if you think of Maxwell's equations.

The potentials come into play later if you want to simplify the equations and find solutions using i.e. Green's functions etc. However, in quantum electrodynamics, the potentials acquire a real physical role, see e.g. the Aharanov-Bohm-effect.

• How does this address "...but using [a] least action principle?" Feb 4, 2013 at 15:24
• Thanks, @RobertFilter. However I think that sigificance of Aharonov-Bohm effect is overstated. All speculations about the physical role of potentials are only valid if particles considered point-like. Feb 4, 2013 at 15:26
• The coupling to matter explicitly involves the potentials. Feb 4, 2013 at 18:17
• @JerrySchirmer This is true, if we write the interaction term in the Lagrangian as the usual $\partial_\mu A^\mu$. However, if you come back to Maxwells equations and Lorentz force, $\propto j_\nu F^{\mu \nu}$, the potentials are not included in any equations of motion in CED. Btw: Who made my comment an answer?! Feb 5, 2013 at 8:57
• @RobertFilter: Maxwell's equations and the Lorentz force aren't part of the action principle. The lagrangian is $\frac{1}{4}F_{ab}F^{ab} + A_{\mu}j^{\mu}$. Most matter lagrangians will break the gauge invariance, so this is not trivial. Feb 10, 2013 at 2:36


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# If trapdoor OWF exists then f is a trapdoor OWF, is there such a construction?

Is there a known construction of f, such that given that a trapdoor OWF exists then f is a trapdoor OWF, so we can construct inefficient cryptomania, ala Levin's construction for minicrypt in "The Tale of One-way functions"? Is there such a construction, which would yield OWF if we are in minicrypt, but TOWF if we are in cryptomania?

format: plain


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# Will JWST be as durable as the hubble telescope?

JW Space Telescope is designed for a 5-10 year mission duration. Seeing as Hubble and other space missions have paved the way for JWST for reliability issues, It even has zero friction gyro's with 100+ years mean time before failure. Is it reasonable to think that JWST can last longer than Hubble?

JWST will orbit a Lagrange point in earth's shadow and will be 4 times further than the moon, around 10 days of travel away, and NASA states that JWST won't be serviceable. Why?

Why are JWST's fuel limitations so short? Are there no provisions for refueling? even a refueling craft? How many gallons of dinitrogen tetroxide ($N_2O_4$) as oxidizer and hydrazine ($N_2H_4$) will it contain?

Surely NASA are secretly hoping for a 20 year mission?

• The JamesWebb hardware will probably remain operational for approximately a zillion years :-). The usual weight vs. lifetime rules apply to the amount of propellant sent out on this mission. – Carl Witthoft Dec 9 '16 at 12:23
• Note that 'no plan for refueling' isn't equivalent to 'cannot become economical within 20 years or less'. – user2338816 Dec 9 '16 at 13:49

Hubble was in low earth orbit, and was always intended to be serviceable. In fact, the original plan for Hubble was to have the space shuttle carry it down from orbit and take it back up, but they decided that was too risky compared to servicing in orbit.

JWST, on the other hand, will be at the Earth/Sun L2 Lagrange point, like WMAP and Planck before it. The distance to that point is 0.01 astronomical units, which is about 1.5 million kilometers from Earth's center. The Moon's orbit around the Earth is about $1/4^{\mathrm{th}}$ of that. So, a manned servicing mission would be an unprecedented project (beyond Apollo), and an unmanned mission, also unprecedented, would face automation challenges from the 5 second light travel time (10 second round trip).

Because the L2 Lagrange point is only meta-stable (stable to offsets in two directions, unstable along the Earth-Sun line), JWST will have to expend fuel on station-keeping just to stay there. It needs to stay there because it can only be as cool as it needs to be by using the Earth as a sun-shade - those tennis court sized shades built in are actually to shade JWST from the thermal radiation of Earth & the moon.

Put simply, it would be cheaper to build and launch another JWST than to service the existing one.

• Thanks! If JWST does survive it's intended duration, then they should definitely have planned a refueling craft and some refueling valves to keep it running. It's cheaper to send a bit of fuel out than an entire new telescope. When there's billions of dollars of equipment in space it's good to refuel and to swap the sensors on it. – com.prehensible Dec 9 '16 at 8:02
• @comprehensible there are conceptual plans right now (Elon Musk, for one) for unmanned refueling missions. I suspect going out to L2 will take a little longer. – Carl Witthoft Dec 9 '16 at 12:22
• "It's cheaper to send a bit of fuel out than an entire new telescope" Hmm, I doubt that would be true. Note that it's tremendously cheaper to buy an iPhone ($100), than it is to make an app for an iPhone ($100,000) - if that analogy makes sense. Or, note that it's incredibly cheaper to just buy a new aircraft, than invent and implement a system of in-air refueling. The idea of "refueling something at L2! robotically!" just seems insanely challenging. – Fattie Dec 9 '16 at 12:36
• Japan uses about 400 million to land on an asteroid, take samples and return. The JWST cost 8 billion. iphones are mass produced, so it's diffrenent than the webb telescope. If they had made only one iphone 7, it would have cost 1-5 billion, and the app would have cost 1 thousandth of that. It's also a cost of man hours and labour, if they can refuel it for an 8th of the price, it saves a lot of resources. A PC chip costs \$100 to change round, but to make a single intel core7 processor cost 4 billion for the fab and many times that for the previous generations of intel processors. – com.prehensible Dec 9 '16 at 13:26
• @comprehensible good points all. however the system to automatically refuel something at L2 would be one of the examples of something that is only cheap when it is mass-produced - it would cost alot to develop such a system. add to that it's now a single point of failure for the extended mission that the refuel would enable, and it becomes more difficult still. if this were done, it also wouldn't solve the problem of replacing broken / worn out components other than fuel, although it would make it possible to deliver them. seems sensible to design the mission without relying on refuelling. – jammypeach Dec 9 '16 at 15:09


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## Delta G = 0

$\Delta G^{\circ}= \Delta H^{\circ} - T \Delta S^{\circ}$

$\Delta G^{\circ}= -RT\ln K$

$\Delta G^{\circ}= \sum \Delta G_{f}^{\circ}(products) - \sum \Delta G_{f}^{\circ}(reactants)$

Helen Mejia 1I
Posts: 67
Joined: Fri Sep 28, 2018 12:24 am

### Delta G = 0

What does it mean when delta G is equal to zero? Does it have to do something with the amount of work the reaction can do?

Rachel-Weisz3C
Posts: 66
Joined: Fri Sep 28, 2018 12:23 am

### Re: Delta G = 0

I believe that if $\Delta$G is equal to zero the reaction is at equilibrium.

Brandon_Tran_2E
Posts: 63
Joined: Fri Sep 28, 2018 12:23 am

### Re: Delta G = 0

Yes!

Rian Montagh 2K
Posts: 60
Joined: Fri Sep 28, 2018 12:15 am

### Re: Delta G = 0

we also set delta G to 0 when we want to find the minimum temp to make a reaction spontaneous.

Also, a reaction at equilibrium does no work, so at delta G = 0 the reaction can do no work.

Cole Elsner 2J
Posts: 88
Joined: Fri Sep 28, 2018 12:25 am

### Re: Delta G = 0

Reaction is at physical equilibrium, generally =0 at phase changes!

Return to “Gibbs Free Energy Concepts and Calculations”

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Synopsis: Two New Particles Enter the Fold

Researchers report the detection of two new subatomic particles made of three quarks from each of the possible quark families.

Physicists at the Large Hadron Collider (LHC) have detected two never-before-seen subatomic particles. These ${\mathrm{\Xi }}_{b}$ particles were predicted by the quark model and estimated to have masses roughly six times that of the proton, but previous experiments have not run at high enough energy to produce these massive particles. The detections offer new precision measurements of the ${\mathrm{\Xi }}_{b}$ masses, which will place tighter constraints on particle physics theories.

Quarks come in three families: up/down, strange/charm, and bottom/top. In 2007, physicists observed the first particle with one quark from each family: ${\mathrm{\Xi }}_{b}^{-}$, consisting of one bottom, one strange, and one down quark, giving it a negative charge of $-1$. However, this is just the lowest mass version of this three-family quark combination. Quark theory predicts the existence of two higher mass cousins of ${\mathrm{\Xi }}_{b}^{-}$, called ${\mathrm{\Xi }}_{b}^{\prime -}$ and ${\mathrm{\Xi }}_{b}^{*-}$, which are characterized by their spin of 1/2 and 3/2, respectively.

To confirm the existence of these short-lived ${\mathrm{\Xi }}_{b}^{-}$ particles, the LHCb experiment at CERN looked for evidence of ${\mathrm{\Xi }}_{b}^{-}$ decays in data from proton-proton collisions at energies of 7 and 8 tera-electron-volts. Specifically, they investigated decays into a neutral ${\mathrm{\Xi }}_{b}^{0}$ and a negatively charged pion (${\pi }^{-}$). They observed signatures for two particles at masses of 5935 and 5955 mega-electron-volts, corresponding to ${\mathrm{\Xi }}_{b}^{\prime -}$ and ${\mathrm{\Xi }}_{b}^{*-}$. The results came as a surprise, as many models predicted that the ${\mathrm{\Xi }}_{b}^{\prime -}$ was not massive enough to decay through this route, and a search at another CERN experiment had not found the equivalent decay of a closely related particle ${\mathrm{\Xi }}_{b}^{\prime 0}$. Using the new, very precise mass measurements, theorists will be able to improve their models—specifically those that predict the mass of other quark-based particles.

This research is published in Physical Review Letters.

–Michael Schirber

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# Activation Energy Formula

Activation energy is the least possible energy required to start a  chemical reaction. The molecules need some kinetic energy or velocity to collide with other molecules to start a reaction. No reaction will take place if the collision doesn’t happen, or molecules don’t have enough kinetic energy. The energy needs to initiate the reaction is known as Activation energy.

The activation energy (Ea) of a reaction is measured in joules(J),  kilojoules per mole(kJ/mol) or kilocalories per mole (kcal/mol)

## Activation Energy Formula

If we know the rate constant k1 and k2 at T1 and T2 the activation energy formula is

Where

$log\frac{k_{2}}{k_{1}}=\frac{E_{a}}{2.303R}\left [ \frac{T_{2}-T_{1}}{T_{1}T_{2}} \right ]$

k1,k2 = the reaction rate constant at T1 and T2

Ea = activation energy of the reaction

R = ideal gas constant=8.3145 J/K·mol

T1,T2 = Absolute Temperature in Kelvin

## Solved Examples

Question 1: The rate of a reaction quadruples when the temperature changes from 290 to 330K. Find the energy of activation of the reaction assuming that it does not change with temperature.

Solution:

Given:

T1 = 290K

T2 = 330K

K2 =4K1

From the Arrhenius equation, we obtain

$log\frac{k_{2}}{k_{1}}=\frac{E_{a}}{2.303R}\left [ \frac{T_{2}-T_{1}}{T_{1}T_{2}} \right ]$

by substituting all the values we get,

$0.6021=\frac{E_{a}}{2.303R}\left [ \frac{40}{(290)(330)} \right ]$

Ea = 1,103276.8/40

=27,581.9 J/mol

Ea = 27.5819 kJ/mol


text:
# Publications

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This paper describes and illustrates functionality of the spNNGP R package. The package provides a suite of spatial regression models …

### Beyond counts and averages: Relating geodiversity to dimensions of biodiversity

Abstract Aim We may be able to buffer biodiversity against the effects of ongoing climate change by prioritizing the protection of …

### Bayesian spatially varying coefficient models in the spBayes R package

This paper describes and illustrates new functionality for fitting spatially varying coefficients models in the spBayes (version 0.4–2) …

### A Case Study Competition Among Methods for Analyzing Large Spatial Data

The Gaussian process is an indispensable tool for spatial data analysts. The onset of the big data” era, however, has lead to …

### Hierarchical Bayesian models for small area estimation of forest variables using LiDAR

Light detection and ranging (LiDAR) data have become almost ubiquitous as a remote sensing tool in forestry estimation and mapping …

### spBayes for Large Univariate and Multivariate Point-Referenced Spatio-Temporal Data Models

In this paper we detail the reformulation and rewrite of core functions in the spBayes R package. These efforts have focused on …


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# Homework Help: Poynting vector of current carrying wire

1. Dec 30, 2017

### Physgeek64

1. The problem statement, all variables and given/known data
A long straight wire of radius a and resistance per unit length R carries a constant current I. Find the Poynting vector N = E × H at the surface of the wire and give
a sketch showing the directions of the current, the electric field E, the magnetic field
H, and N. Integrate the Poynting vector over the surface of a length l of the wire and comment on your result

2. Relevant equations

3. The attempt at a solution
$\integral{H\dot dl}=I_{free}$
$H(2\pi r)=I$
$H|_a=\frac{I}{2\pi a} \hat{\theta}$

E outside the wire will be $E=-\frac{\lambda}{2\pi\epsilon r } \hat{r}$

I don't know if this is the same as the E field at the surface? Also I don't know how to get $\lambda$ in terms of I and R

then $S=E\cross H$

Many thanks

2. Dec 30, 2017

### kuruman

Does a wire piece of length $l$ carry a net charge or is it neutral?

3. Dec 30, 2017

### Physgeek64

It is neutral. However I know there needs to be a poynting vector so I don't know how this works

4. Dec 30, 2017

### kuruman

If it's neutral, there is no E-field outside the wire. How about inside the wire?

5. Dec 31, 2017

### Physgeek64

Inside the wire we have a non-neutral charge. Negative electrons flow in one direction, leaving positive ions behind. Hence we have an E-field in the direction of current? Not sure how to calculate its magnitude though

6. Dec 31, 2017

### kuruman

You know that the potential difference across the two ends of the wire is V. If the distance between the ends is $l$, what is E?

7. Dec 31, 2017

### Physgeek64

$\frac{V}{l}$

8. Dec 31, 2017


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# Testing - unit tests¶

The Ceph GitHub repository has two types of tests: unit tests (also called make check tests) and integration tests. Strictly speaking, the make check tests are not “unit tests”, but rather tests that can be run easily on a single build machine after compiling Ceph from source, whereas integration tests require package installation and multi-machine clusters to run.

## What does “make check” mean?¶

After compiling Ceph, the code can be run through a battery of tests For historical reasons, this is often referred to as make check even though the actual command used to run the tests is now ctest. For inclusion in this group of tests, a test must:

• bind ports that do not conflict with other tests

• not require root access

• not require more than one machine to run

• complete within a few minutes

For the sake of simplicity, this class of tests is referred to as “make check tests” or “unit tests”. This is meant to distinguish these tests from the more complex “integration tests” that are run via the teuthology framework.

While it is possible to run ctest directly, it can be tricky to correctly set up your environment. Fortunately, a script is provided to make it easier run the unit tests on your code. It can be run from the top-level directory of the Ceph source tree by invoking:

.. prompt:: bash $You will need a minimum of 8GB of RAM and 32GB of free drive space for this command to complete successfully on x86_64; other architectures may have different requirements. Depending on your hardware, it can take from twenty minutes to three hours to complete, but it’s worth the wait. ## How unit tests are declared¶ Unit tests are declared in the CMakeLists.txt file, which is found in the ./src directory. The add_ceph_test and add_ceph_unittest CMake functions are used to declare unit tests. add_ceph_test and add_ceph_unittest are themselves defined in ./cmake/modules/AddCephTest.cmake. Some unit tests are scripts and other unit tests are binaries that are compiled during the build process. • add_ceph_test function - used to declare unit test scripts • add_ceph_unittest function - used for unit test binaries ## Unit testing of CLI tools¶ Some of the CLI tools are tested using special files ending with the extension .t and stored under ./src/test/cli. These tests are run using a tool called cram via a shell script ./src/test/run-cli-tests. cram tests that are not suitable for make check may also be run by teuthology using the cram task. ### Tox based testing of python modules¶ Most python modules can be found under ./src/pybind/. Many modules use tox to run their unit tests. tox itself is a generic virtualenv management and test command line tool. To find out quickly if tox can be run you can either just try to run tox or check for the existence of a tox.ini file. Currently the following modules use tox: • Cephadm (./src/pybind/mgr/cephadm) • Insights (./src/pybind/mgr/insights) • Manager core (./src/pybind/mgr) • Dashboard (./src/pybind/mgr/dashboard) • Python common (./src/python-common/tox.ini) Most tox configurations support multiple environments and tasks. You can see which are supported by examining the envlist assignment within tox.ini To run tox, just execute tox in the directory where tox.ini is found. If no environments are specified with e.g. -e$env1,$env2, all environments will be run. Jenkins will run tox by executing run_tox.sh which is under ./src/script. Here some examples from the Ceph Dashboard on how to specify environments and run options: ## Run Python 2+3 tests+lint commands:$ tox -e py27,py3,lint,check

## Run Python 3 tests+lint commands:
$tox -e py3,lint,check ## To run it like Jenkins would do$ ../../../script/run_tox.sh --tox-env py27,py3,lint,check
\$ ../../../script/run_tox.sh --tox-env py3,lint,check

#### Manager core unit tests¶

Currently only doctests inside mgr_util.py are run.

To add test additional files inside the core of the manager, add them at the end of the line that includes mgr_util.py within tox.ini.

## Unit test caveats¶

1. Unlike the various Ceph daemons and ceph-fuse, unit tests are linked against the default memory allocator (glibc) unless explicitly linked against something else. This enables tools like valgrind to be used in the tests.


text:
• # question_answer Which of the following may exist in enantiomorphs [CBSE PMT 1988] A) $C{{H}_{3}}-\overset{C{{H}_{3}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}{\mathop{\overset{|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}{\mathop{CH-COOH\,}}\,}}\,$ B) $C{{H}_{2}}=CHC{{H}_{2}}C{{H}_{2}}C{{H}_{3}}$ C) $C{{H}_{3}}-\overset{N{{H}_{2}}\,\,\,\,\,\,\,\,\,\,\,\,}{\mathop{\overset{|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}{\mathop{CH-C{{H}_{3}}\,}}\,}}\,$ D) $C{{H}_{3}}-C{{H}_{2}}-\overset{N{{H}_{2}}\,\,\,\,\,\,\,\,\,\,\,\,}{\mathop{\overset{|\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,}{\mathop{CH-C{{H}_{3}}}}\,}}\,$

$C{{H}_{3}}-C{{H}_{2}}-\underset{H\,\,\,\,\,\,\,\,\,}{\overset{N{{H}_{2}}}{\mathop{\underset{|\,\,\,\,\,\,\,\,\,\,\,}{\overset{|\,\,\,\,\,\,\,\,\,\,}{\mathop{{{C}^{*}}\,\,\,-\,}}}\,}}}\,C{{H}_{3}}$ Chiral centre is present. Hence, it exists as optical isomers or enantiomorphs.


text:
## 2.10.2012

### Wish List 2.10.12

Last night, I did some experimenting with Photoshop on the train. I've been having some arrow frustrations (yes, that's a real thing!). The Photoshop arrows just... well, they suck and look like kindergarten decor (no offense, Mom). I've been dreaming of some pencil/chalk-looking swirly arrows that I could free hand in Photoshop instead of using a pre-defined arrow shape (because, let's face it, pre-defined arrows are just less fancy).

Anyways (stop snoring!), here's what I ended up with! For the Photoshop junkies out there, I'll be back in the next few days with a tutorial on those fabulously chalk-and-pencil-ified arrows!

For now, here are my three four wishes...

Pink Wayfarers from Woodzee
Chicago Skyline Wallart Mural
Knappa Lighting
Boho Fringe Bag


text:
# What is FrontEndAttachedCell?

Perhaps it's not accessible from the front end, but what is this so-called attached cell?

In[91]:= Names["**AttachedCell*"]

Out[91]= {"FrontEndAttachedCellParent", \
"PredictiveInterfaceDump$UseAttachedCells", \ "WebPredictionsPrivate$UseAttachedCells"}

-
The title of your question does not seem to be related in any way to body of your question. The title is too broad to be a good question. The body is obscure; what motivated you to search on "**AttachedCell*"? – m_goldberg Nov 13 '15 at 4:15
Oh I was just guessing from some spelunking that attachedcells are the mechanism it uses. – M.R. Nov 13 '15 at 4:23
@m_goldberg fixed the title. – Kuba Nov 13 '15 at 9:10

## Intro

One day I was playing with Developer package and found DateSetter which uses some kind of floating elements that I found useful.

After taking a look at a source code I found out it was FrontEndAttachedCell. From the code one could learn enough to create something useful:

## Usage

FrontEndAttachCell[
parentObject,             (*Box or Cell Object*)
attachedCellBoxes,
{Automatic, {Left, Bottom}}, (*1*)
{Left, Top},                 (*2*)
"ClosingActions" -> {...}    (*3*)
]

1. I don't know what an Automatic is there for but the second part is setting alignment anchor point in the parentBoxObject
2. alignment anchor point in the attached cell
3. "The ClosingActions determines actions which automatically dismiss the attached cells". Available actions: "ParentChanged", "EvaluatorQuit", "OutsideMouseClick", "SelectionDeparture", "MouseExit".

I couldn't figure everything out by myself so let's quote John Fultz whom I've asked about details:

• Kuba:

[...]

1. I'm aware it is undocumented but I'm curious if it is stable. [...]

2. Can you give me any insight in "ClosingActions"? [...]

3. Can I AttachCell for DockedCells elements? [...]

• John Fultz:

1. Yes, this functionality is used all over the place. At some point, it'll graduate to a System level function, but even when it does, we'll probably maintain backward compatibility with the current stuff.

2. The "ClosingActions" determines actions which automatically dismiss the attached cells. You can manually dismiss a cell by doing NotebookDelete on the attached cell's CellObject. But there are some circumstances where it's much easier to let the system do it for you. It can be a list of any of the following:

"SelectionDeparture" -- if the selection leaves the parent, the attached cell, or any of the attached cell's children (the selection might be in the attached cell if the attached cell has InputFields).

"ParentChanged" -- if any change is made to the parent of the attached cell

"MouseExit" -- if the mouse leaves the region of the attached cell

"OutsideMouseClick" -- if a click happens anywhere outside of the attached cell. This includes clicks of the right and middle mouse buttons.

1. Yes, that works, too. Except that the docked view will clip attached cells, so if the attached cell doesn't live fully within the docked view, it may not be very useful.

## My example:

attachTo[parentbox_] := MathLinkCallFrontEnd[
FrontEndAttachCell[
parentbox,
ToBoxes[ExpressionCell[
EventHandler[Panel["floating panel"],
"MouseExited" :> (NotebookDelete[ParentCell[EvaluationBox[]]];)
],
StripOnInput -> True, Background -> White,
CellFrameColor -> LightBlue, CellFrameMargins -> 0, CellFrame -> 2
]],
{Automatic, {Right, Bottom}},
{Left, Top},
"ClosingActions" -> {"ParentChanged", "EvaluatorQuit",  "OutsideMouseClick"}
]]

DynamicModule[{parentBox, attachedCell},
EventHandler[
Panel["test"],
{ "MouseEntered" :> (
NotebookDelete @ attachedCell;
attachedCell = attachTo[parentBox];
)}
],
Initialization :> ( parentBox = EvaluationBox[]; )
]

Notice that I'm not using "MouseExit" closing action but an EventHandler + NotebookDelete. That's because if the initial position of the attached cell is away from cursor then it won't appear at all.

## Possible issues

• the tricky thing is to get the parentBoxObject.
• it's not the same as e.g. menu in PopupMenu since attached cell is restricted to notebook area only while the menu is able to be outside.
-
Thank you very much for the canonical answer (as always) :) – M.R. Nov 13 '15 at 21:34
@M.R. You are welcome, and thanks for another interesting question :) – Kuba Nov 13 '15 at 22:00
+1 Thanks for the digging Kuba! I was waiting for this kind of interactive floating element for years now. Didn't realize that DateSetter had it. – István Zachar Nov 14 '15 at 9:21
@IstvánZachar No problem, I have just recently found it too. I hope this post will generate many topics with nice GUI based on this. – Kuba Nov 14 '15 at 9:25
Aren't you missing an assignment as attachedCell = attachTo[parentBox] within the parent's EventHandler? – István Zachar Dec 13 '15 at 16:25


text:
# Eigenvalue/function of Laplace Operator for Exterior of Disk - Reference Request

I'd like to know more about,

$$\lambda u = \nabla ^2 u$$

for unbounded domains (particularly the exterior of a disk in $$\mathbb{R}^2$$ or ball in $$\mathbb{R}^3$$), but have had a hard time finding references online. Can anyone suggest a website or book that covers this in depth? i.e. How to obtain solutions, proofs, important corollaries/theorems, etc.

• Have you tried separation of variablaes? – DisintegratingByParts Oct 2 '18 at 22:02
• Facepalm... will try that first.... although, the reason I bring it up is because I recall that unbounded domains can dramatically alter the nature of the eigenvalues. But first things first. – Eric Oct 2 '18 at 22:25
• It's natural to restrict to $L^2$ solutions on the exterior domain. – DisintegratingByParts Oct 4 '18 at 2:36


text:
## Robots 101: Lasers

April 29, 2015

By Ilia Baranov

In this new Robots 101 series, we will be taking a look at how robots work, what makes designing them challenging, and how engineers at Clearpath Robotics tackle these problems. To successfully operate in the real world, robots need to be able to see obstacles around them, and locate themselves. Humans do this mostly through sight, whereas robots can use any number of sensors. Today we will be looking at lasers, and how they contribute to robotic systems.

## Overview

When you think of robots and lasers, the first image that comes to mind might come from science fiction; robots using laser weapons. However, almost all robots today use lasers for remote sensing. This means that the robot is able to tell, from a distance, some characteristics of an object. This can include size, reflective, color, etc. When the laser is used to measure distance in an arc around the robot, it is called LIDAR. LIDAR is a portmanteau of Light and Radar, essentially think of a sweeping radar beam shown in the films, using light.

Function and Concepts

All LIDAR units operate using this basic set of steps:

1. Laser light is emitted from the unit (usually infrared)
2. Laser light hits an object and is scattered
3. Some of the light makes it back to the emitter
4. The emitter measures the distance (more on how later)
5. Emitter turns, and process begins again

A great visual deception of that process to the right.

## Types of LIDAR sensing

How exactly the laser sensor measures the distance to an object depends on how accurate the data needs to be. Three different methods commonly found on LIDAR sensors are:

Time of flight
The laser is emitted, and then received. The time difference is measured, and distance is simply (speed of light) x (time). This approach is very accurate, but also expensive due to the extremely high precision clocks needed on the sensor. Thus, it is usually only used on larger systems, and at longer ranges. Rare to use on robots

Phase difference

In this method, the emitted laser beam is modulated to a specific frequency. By comparing the phase shift of the returned signal on a few different frequencies, the distance is calculated. This is the most common way laser measurement is done. However, it tends to have limited range, and is less accurate than time of flight. Almost all of our robots use this.

Angle of incidence
The cheapest and easiest way to do laser range-finding is by angle of incidence. Essentially, by knowing what angle the reflected laser light hits the sensor, we can estimate the distance. However, this method tends to be of low quality. The Neato line of robotic vacuum cleaners use this technology.

Another thing to note is that so far, we have only discussed 2D LIDAR sensors. This means they are able to see a planar slice of the world around them, but not above or below. This has limitations, as the robot is then unable to see obstacles lower or higher than that 2d plane. To fix this issue, either multiple laser sensors can be used, or the laser sensor can be rotated or “nodded” up and down to take multiple scans. The first solution is what the Velodyne line of 3D LIDAR sensors employ, while the second solution tends to be a hack done by roboticists. The issue with nodding a LIDAR unit is a drastic reduction of refresh rate, from tens of Hz, down to 1 Hz or less.

## Manufacturers and Uses

Some of the manufacturers we tend to use are SICK, Hokuyo, and Velodyne.
This is by no means an exhaustive list, just the ones we use most often.

SICK
Most of our LIDARs are made by SICK. Good combo of price, size, rugged build, and software support. Based in Germany

Hokuyo
Generally considered cheaper than SICK, in larger quantities. Some issues with fragility, and software support is not great. Based in Japan

Velodyne
Most expensive, used only when robot MUST see everything in the environment. Based in US.

Once the data is collected, it can be used for a variety of reasons. For robots, these tend to be:

• Mapping
• Use the laser data to find out dimensions and locations of obstacles and rooms. This allows the robot to find its position (localisation) and also report dimensions of objects. The title image of this article shows the Clearpath Robotics Jackal mapping a room.
• Survey
• Similar to mapping, however this tends to be outside. The robot collects long range data on geological formations, lakes, etc. This data can then be used to create accurate maps, or plan out mining operations.
• Obstacle Avoidance
• Once an area is mapped, the robot can navigate autonomously around it. However, obstacles that were not mapped (for example, squishy, movable humans) need to be avoided.
• Safety Sensors
• In cases where the robot is very fast or heavy, the sensors can be configured to automatically cut out motor power if the robot gets too close to people. Usually, this is a completely hardware-based feature.

## Selection Criteria

A lot of criteria is used to select the best sensor for a given application. The saying goes that an engineer is someone who can do for $1 what any fool can do for$2. Selecting the right sensor for the job not only reduces cost, but also ensures that the robot has the most flexible and useful data collection system.

• Range
• How far can laser sensor see? This impacts how fast a robot is able to move, and how well it is able to plan.
• Light Sensitivity
• Can the laser work properly outdoors in full sunlight? Can it work with other lasers shining at it?
• Angular Resolution
• What is the resolution of the sensor? More angular resolution means more chances of seeing small objects.
• Field of view
• What is the field of view? A greater field of view provides more data.
• Refresh rate
• How long does it take the sensor to return to the same point? The faster the sensor is, the faster the robot can safely move.
• Accuracy
• How much noise is in the readings? How much does it change due to different materials?
• Size
• Physical dimensions, but also mounting hardware, connectors, etc
• Cost
• What fits the budget?
• Communication
• USB? Ethernet? Proprietary communication?
• Power
• What voltage and current is needed to make it work?
• Mechanical (strength, IP rating)
• Where is the sensor going to work? Outdoors? In a machine shop?
• Safety
• E-stops
• regulation requirements
• software vs. hardware safety

For example, here is a collected spec sheet of the SICK TIM551.

 Range (m) 0.05 – 10 (8 with reflectivity below 10%) Field of View (degrees) 270 Angular Resolution (degrees) 1 Scanning Speed (Hz) 15 Range Accuracy (mm) ±60 Spot Diameter (mm) 220 at 10m Wavelength (nm) 850 Voltage (V) 10 – 28 Power (W) (nominal/max) 3 Weight (lb/kg) 0.55/0.25 Durability (poor) 1 – 5 (great) 4 (It is IP67, metal casing) Output Interface Ethernet, USB (non-IP67) Cost (USD) ~2,000 Light sensitivity This sensor can only be used indoors. Other Can synchronize for multiple sensors.Connector mount rotates nicely.

Clearpath Robotics Clearpath Robotics is dedicated to automating the world's dullest, dirtiest and deadliest jobs through mobile robotic solutions. .. read more

### podcast

ICRA 2018 Exhibition
June 23, 2018

Are you planning to crowdfund your robot startup?


text:
There exists a regular language A such that for all languages B, A ∩ B is regular.

There exists a regular language A such that for all languages B, A ∩ B is regular.

The above given statement is true but I couldn't make any proof or find any proof. It is an objective type question asked here to find whether the given statement is true or false. I want to know how to conclude this given statement is true.

• There exists infinite regular languages $A$ such that for all languages $B$, $A \cap B$ is regular. Take any finite $A \subseteq \Sigma^*$. – Bakuriu Jul 16 '16 at 20:06
• @Bakuriu You probably meant the opposite of your first sentence. – 6005 Jul 17 '16 at 0:16
• @6005 It depends on how you parse that sentence. It should be read as $\exists^{\infty} A ....$ (there exist an infinite number of languages $A$ such that) not $\exists A, |A|=\infty ...$ (there exists one language, which is infinite, ..) – Bakuriu Jul 17 '16 at 8:17

Yes, that's true. Consider $A=\emptyset$ (which is regular), then $\emptyset \cap B=\emptyset$ (which is regular).

If $A$ is a finite language, then it is regular and meets your condition.

On the other hand if $A$ is any infinite regular language, since it is countably infinite ($\aleph_0$) it will contain $\aleph_1$ sublanguages. Every regular language is defined by a finite regular expression (of which there are $\aleph_0$) so there will be sublanguages of $A$ which is not regular.

So finiteness is necessary and sufficient.

• Note that your answer depends on the continuum hypothesis. You probably just meant "uncountably many" which is all you need for your argument, but without irrelevant assumptions, you would have $2^{\aleph_0}$ sublanguages. – Richard Rast Jul 16 '16 at 15:42
• Rule of thumb: Unless you're playing around with the CH or the ordinals, you want either beth numbers or some expression involving $\aleph_0$. – Kevin Jul 16 '16 at 16:27
• @Richard: Or simply ‘at least $\aleph_1$’, though I too would prefer ‘uncountably many’ here, since that gets directly to the essential point. – Brian M. Scott Jul 16 '16 at 20:14
• As a constructive way to make a nonregular sublanguage, take all the sentences, order them by length, then take a subsequence where the length grows too quickly to be regular. – Owen Jul 16 '16 at 22:08
• @YBerman That sounds right to me. – Owen Jul 17 '16 at 11:32


text:
# Energy profile of Nigeria

May 10, 2013, 6:54 pm
 Topics:

Nigeria is the largest oil producer in Africa and has been a member of the Organization of Petroleum Exporting Countries (OPEC) since 1971. In 2011, Nigeria produced about 2.53 million barrels per day (bbl/d) of total liquids, well below its oil production capacity of over 3 million bbl/d, due to production disruptions that have compromised portions of the country's oil for years.

 Nigeria's hydrocarbon resources are the mainstay of the country's economy, but development of the oil and natural gas sectors is often constrained by instability in the Niger Delta.

The Nigerian economy is heavily dependent on its hydrocarbon sector, which accounted for more than 95 percent of export earnings and more than 75 percent of federal government revenue in 2011, according to the International Monetary Fund (IMF).

The oil industry is primarily located in the Niger Delta where it has been a source of conflict. Local groups seeking a share of the oil wealth often attack the oil infrastructure and staff, forcing companies to declare force majeure on oil shipments. At the same time, oil theft, commonly referred to as "bunkering," leads to pipeline damage that is often severe, causing loss of production, pollution, and forcing companies to shut-in production. Protest from local groups over environmental damages from oil spills and flaring undermined relations between local communities and international oil companies (IOCs). The industry has been blamed for pollution that has damaged air, soil, and water, leading to losses in arable land and decreasing fish stocks.

In addition to oil, Nigeria holds the largest natural gas reserves in Africa, but has limited infrastructure in place to develop the sector. Natural gas that is associated with oil production is mostly flared, but the development of regional pipelines, the expansion of liquefied natural gas (LNG) infrastructure, and policies to ban gas flaring are expected to accelerate growth in the sector, both for export and domestic use in electricity generation. Uncertainties in Nigeria's investment policies and regulatory framework have caused a slowdown in oil and gas exploration activity, and delays in project development, including LNG projects. However, the long-awaited and delayed Petroleum Industry Bill (PIB) could potentially iron out investment uncertainties and set a regulatory framework for the country's oil and gas industry.

The first draft of the PIB was initially introduced in 2008, with the purpose of restructuring the hydrocarbon sector, clarifying regulatory and operational roles of Nigerian energy institutions, and increasing government take and local content requirements. Passage of the PIB has been stalled by a lack of support, notably by IOCs, and also ongoing debate within the Nigerian government. Nonetheless, after several rounds of revisions, there are indications that IOCs have a more positive perception of the bill, although concerns have been expressed, most recently by Shell. The PIB has been sent to Nigeria's National Assembly.

EIA estimates that in 2010 total energy consumption was about 4.4 Quadrillion Btu (111,000 kilotons of oil equivalent). Of this, traditional biomass and waste accounted for 82 percent of total energy consumption. This high percent share represents the use of biomass to meet off-grid heating and cooking needs, mainly in rural areas. IEA data for 2009 indicate that electrification rates for Nigeria were 50 percent for the country as a whole – leaving approximately 76 million people without access to electricity in Nigeria. Other estimates place the countrywide electrification rate as low as 45 percent.

Nigeria has vast natural gas, coal, and renewable energy resources that could be used for domestic electricity generation. However, the country lacks policies to harness resources and develop new (and improve current) electricity infrastructure. The Nigerian government has had several plans to address the need for power, including a recent announcement to create 40 gigawatts (GW) of capacity by 2020 (compared to 2009 installed capacity of 6 GW). Achieving this goal will mainly depend on the ability of the Nigerian government to utilize currently flared natural gas.

# Oil

For the last nine years, the U.S. has imported between 9-11 percent of its crude oil from Nigeria; however, U.S. import data for the first half of 2012 show that Nigerian crude is down to a 5 percent share of total U.S. crude imports.

According to Oil and Gas Journal (OGJ), Nigeria has an estimated 37.2 billion barrels of proven oil reserves as of the end of 2011. The majority of reserves are found along the country's Niger River Delta and offshore in the Bight of Benin, the Gulf of Guinea, and the Bight of Bonny. Current exploration activities are mostly focused in the deep and ultra-deep offshore with some activities in the Chad basin, located in the northeast of the country.

The government hopes to increase proven oil reserves to 40 billion barrels in the next few years; however, exploration activity levels are at their lowest in a decade and only three exploratory wells were drilled in 2011, compared to over 20 in 2005. Rising security problems related to oil theft, pipeline sabotage, and piracy in the Gulf of Guinea, coupled with investment uncertainties surrounding the long-delayed PIB, have curtailed oil exploration projects and impeded the country from reaching its ongoing target to increase production to 4 million bbl/d. Instead, crude oil production averaged 2.13 million bbl/d in 2011, roughly the same as it was a decade ago, and total liquids production averaged 2.53 million that same year, which is still below the peak production of 2.63 million bbl/d reached in 2005.

## Production

In 2011, crude oil production averaged close to 2.13 million bbl/d, up from 2.05 million bbl/d in the previous year. EIA's recent estimates show that crude output rose slightly again in 2012 and averaged almost 2.15 million bbl/d for the first half of this year. The recent increase in production is due to the expansion of existing fields and new production from deepwater fields. The latest major deepwater field to come onstream was Total's Usan field, which began producing over 100,000 bbl/d in July 2012 and is expected to reach 180,000 bbl/d by the end of this year.

Oil production in Nigeria reached its peak of 2.63 million bbl/d in 2005, but began to decline significantly as violence from militant groups surged, forcing many companies to withdraw staff and shut in production. The lack of transparency of oil revenues, tensions over revenue distribution, and environmental damages from oil spills, coupled with local ethnic and religious tensions, have created a fragile situation in the oil-rich Niger Delta basin. As a result, crude oil production plummeted by more than 25 percent by 2009, four years after reaching its peak.

Towards the end of 2009, an amnesty was declared and the militants came to an agreement with the government whereby they handed over weapons in exchange for cash payments and training opportunities. The rise in oil production after 2009 was partially due to the reduction in attacks on oil facilities following the implementation of the amnesty program, which allowed companies to repair some damaged infrastructure and bring some supplies back online. Another major factor that contributed to the upward trend in output was the continued increase in new deepwater offshore production. The government began taking measures to attract investment in deepwater acreage in the 1990s in order to boost production capacity and diversify the country's oil fields, as security issues in the Niger Delta escalated. In order to incentivize investments in deepwater areas, which involve higher capital and operating costs, the government offered production-sharing contracts (PSC) in which IOCs received a greater share of revenue as the depth increased.

Although terms within the PSCs have been revised over time to provide the government with larger shares in revenue, the policy did facilitate greater investment and production in deepwater fields. The first deepwater field began production in 2003, and since then output from deepwater fields has added over 800,000 bbl/d to the country's production capacity.

As an OPEC member, Nigeria has agreed to a crude oil production quota of 1.704 million bbl/d. However, the country still plans on bringing online several projects in the next few years. Planned upstream developments, particularly deepwater projects, should increase Nigerian oil production in the medium term, but the timing of these startups will depend heavily on the passing of the PIB and the fiscal/regulatory terms it requires of the oil industry. Many of the planned projects described below have already been delayed.

Upcoming oil projects in Nigeria
Project Capacity
('000 bbl/d)
Est. Startup Sector Operator

1Expansion of existing Agbami field- drilling activities expected to continue through 2014 (Chevron).
2Production began in 2010 and is expected to ramp up to 70,000 bbl/d once all wells are drilled.
3Ofon (phase 1) is currently producing around 30,000 bbl/d and phase 2 is expected to increase capacity to 90,000 bbl/d.
Note: Deepwater projects have a water depth greater than 200 meters.
Sources: Oil and Gas Journal; IEA Medium Term Oil Market Report; IHS Cera, Wood Mackenzie; Total; Chevron; Rigzone; Business Week; OPEC Secretariat

Agbami1 100 2011-2014 Deepwater Chevron
Ebok (phase 2) 35 2012 Offshore Afren
Gbaran Ubie2 70 2012+ Onshore Shell
Ehra North (phase 2) 50 2013+ Deepwater ExxonMobil
Oberan tbd 2013+ Deepwater Eni (Agip)
Ofon (phase 2)3 90 2014 Offshore Total
Aje tbd 2014 Deepwater Yinka Folawyo Petroleum
Bonga North, Northwest 50-150 2014+ Deepwater Shell
Bonga Southwest and Aparo 140 2014+ Deepwater Shell
Egina 150-200 2014+ Deepwater Total
Bosi 135 2015 Deepwater ExxonMobil
Nsiko 100 2015+ Deepwater Chevron
Uge 110 2016 Deepwater ExxonMobil
Nkarika tbd 2019 Offshore Total
Etan/Zabazaba 110 tbd Deepwater Eni (Agip)
Okan 35 tbd Offshore Chevron

#### Security risks

Since December 2005, Nigeria has experienced increased pipeline vandalism, kidnappings, and militant takeovers of oil facilities in the Niger Delta. The Movement for the Emancipation of the Niger Delta (MEND) is the main group attacking oil infrastructure for political objectives, claiming to seek a redistribution of oil wealth and greater local control of the sector. Additionally, kidnappings of oil workers for ransom are common and security concerns have led some oil services firms to pull out of the country and oil workers unions to threaten strikes over security issues. The instability in the Niger Delta has also caused significant amounts of shut-in production at onshore and shallow offshore fields, and forced several companies to declare force majeure on oil shipments.

The amnesty program implemented in 2009 led to decreased attacks in 2009-2010 and some companies were able to repair damaged oil infrastructure. However, the lack of progress in job creation and economic development has led to increased bunkering and other attacks in 2011.

Bunkering, which in the context of Nigeria's oil industry refers to the theft and trade of stolen oil, has recently surged, and according to NNPC data, pipeline vandalism increased by 224 percent in 2011 over the previous year. Estimates from Nigeria's Ministry of Finance show that about 400,000 bbl/d of oil was stolen in April 2012, which led to a fall of about 17 percent in official oil sales. Royal Dutch Shell, Nigeria's largest producer, recently estimated that 150,000-180,000 bbl/d, or 6 percent of the country's total production, on average is lost to oil bunkering and spills.

According to information disseminated by an investigative task force in Nigeria, there are three main ways oil is bunkered: by small cargo canoes that navigate the swampy, shallow waters of the Niger Delta where culprits puncture pipelines to siphon crude into small tanks; stealing crude directly from the wellhead; or filling tankers at export terminals, which is referred to as "white collar" bunkering. Some stolen oil is taken to illegal refineries along the Niger Delta's swampy bush areas and sold domestically and regionally, while other portions make their way to the international market. Some analysts believe that white collar bunkering is not included in Shell's oil theft estimate and that the average amount stolen is actually closer to the Finance Ministry's estimate, although there is no official number.

In addition to losses in official oil sales, oil theft and illegal refineries are causing environmental damages and costing the country $7 billion a year, according to the Nigerian government. According to the Nigerian National Oil Spill Detection and Response Agency (NOSDRA) approximately 2,400 oil spills had been reported between 2006 and 2010 that resulted from sabotage, bunkering, and poor infrastructure. The amount of oil spilled in Nigeria has been estimated to be around 260,000 barrels per year for the past 50 years, according to a report cited in the New York Times. The oil spills have caused land, air, and water pollution and severely affected surrounding villages by decreasing fish stocks and contaminating water supplies and arable land. The United Nations Environment Program (UNEP) released a study on Ogoniland and the extent of environmental damage from over 50 years of oil production in the region. The study confirmed community concerns regarding oil contamination across land and water resources, stating that that the damage is ongoing and estimating that it could take 25 to 30 years to repair. Incidents of piracy in the Gulf of Guinea have posed a risk to deepwater offshore operations. According to the International Maritime Organization (IMO), there were 53 piracy related attacks in the Gulf of Guinea in 2011, up from the 47 in 2010. The attacks typically involve stolen cargo, especially crude oil, and violence against crew members. The country's deepwater offshore production has been relatively unharmed by the country's instability, as oil platforms are miles out from the coast, but piracy does pose a security risk to deepwater offshore production. Although incidences of Nigerian piracy are still less than in Somalia, Nigerian piracy is becoming more frequent and happening at greater distances away from the coast, according to IMO. ## Sector organization In 1977, Nigeria created the Nigerian National Petroleum Company (NNPC). At that time, NNPC's primary function was to oversee the regulation of the Nigerian oil industry, with secondary responsibilities for upstream and downstream developments. In 1988, the Nigerian government divided the NNPC into 12 subsidiary companies in order to better manage the country's oil industry. The majority of Nigeria's major oil and natural gas projects are managed through JVs with the NNPC. #### Recent developments The government has been planning to transform NNPC into a more profit-driven company that can seek out private financing. While these discussions have been underway for many years, a Petroleum Industry Bill (PIB) is currently being debated by the National Assembly to reform the entire hydrocarbon sector. Parts of the bill have recently been approved as standalone laws, while the different agencies and roles of the new national oil company and the NNPC have yet to be fully defined. Differing versions of the PIB are currently under debate, especially around more contentious points such as the renegotiation of contracts with international oil companies, the changes in tax and royalty structures, and clauses to ensure that companies use or lose their assets. The ongoing debate has delayed investments in both the oil and natural gas sectors. As part of the energy sector reform, in April 2010, then acting president (now president) Goodluck Jonathan signed the Nigerian Content Development Bill (NCD) into law. The bill aims to increase the role of Nigerian companies in all aspects of the oil and gas industry. The new law requires that Nigerian companies obtain contracts and win bids so long as the local company is capable, the Nigerian content is higher, and the bid is not more than 10 percent higher than the competing bid. According to the African Oil and Gas Monitor (Afroil), the NCD applies to all contracts worth over US$1 million and also applies to insurance, banking, and other sectors tied into the oil industry.

The distribution of oil revenue has been a very contentious issue in the country, since all revenue, including production proceeds, corporate tax, customs duties, and value-added tax, goes directly to the federal government account. The lack of transparency and mismanagement of oil revenue has sparked mistrust among the federal government, states, and local councils. The 1999 constitution carved out a revenue-sharing arrangement in which 13 percent of oil revenue from onshore production goes directly to the nine oil producing states in the Niger Delta, with the remaining revenue allocated to the federal government (47.2 percent), states (31.1 percent), local councils (15.2 percent), and National Priorities Services Fund (6.5 percent).

Disagreement over the current revenue-sharing arrangement is one of the main issues driving the political tension, theft, and sabotage in the Niger Delta, and groups have demanded to extend their revenue-share to offshore production and increase their onshore revenue-share to 50 percent. However, the effect that the PIB will have on the current revenue-sharing arrangement is unclear.

Petroleum Industry Bill (PIB), draft 2012
Key points

Note: These measures may not appear in the final version of the PIB.
Source: Energy Intelligence, Reuters, Financial Times, and IHS Cera.

Increase exploration activities and expand reserves
Monetize natural gas reserves and reduce flaring
Separate regulators for the upstream, midstream, and downstream sectors
Deregulate the downstream sector
Offer acreage through bid rounds
Increase government take
Higher royalties
Lower production taxes
Increase local participation through employment, related industries, and local oil & gas companies
Petroleum Host Communities Fund (PHCF)

## International oil companies

Foreign companies operating in joint ventures (JVs) or production sharing contracts (PSCs) with the NNPC include ExxonMobil, Chevron, Total, Eni, Addax Petroleum (recently acquired by Sinopec of China), ConocoPhillips, Petrobras, StatoilHydro, and others.

Shell has been working in Nigeria since 1936, and currently operates the largest nameplate crude oil production capacity, estimated to be between 1.2-1.3 million bbl/d. However, the company has been hardest hit by the instability as much of its production is in shallow water and onshore the Niger Delta. Much of Shell's crude oil production capacity is shut-in, some since as far back as early 2006. According to Shell, the total oil produced from Shell-run operations averaged 974,000 bbl/d in 2011.

Shell operates in Nigeria through the Shell Petroleum Development Company of Nigeria Limited (SPDC) and the Shell Nigeria Exploration and Production Company Limited (SNEPCo). SPDC is the largest oil and gas company in Nigeria and is a joint venture between NNPC (55%), Shell (30%), Elf Petroleum Nigeria Limited — a subsidiary of Total — (10%), and Agip (5%). SPDC's operations include a network of pipelines, nine gas plants, and two export terminals. Shell owns 100 percent of SNEPCo, which was formed in 1993 to develop Nigeria's deepwater oil and gas resources offshore. Under a PSC with NNPC, it operates the Bonga deepwater oil and gas project and is a venture partner in the Erha deepwater oil and gas project with ExxonMobil.

ExxonMobil, the second largest IOC, operates fields producing approximately 800,000 bbl/d (700,000 bbl/d of crude) in partnership with NNPC. Chevron is the third largest oil producer in Nigeria and produced an average of 516,000 bbl/d of crude oil in 2011. The company operates under its subsidiary, Chevron Nigeria Limited, and holds 40 percent interest in 13 concessions under a joint venture arrangement with NNPC. Most of its oil projects are in shallow water and onshore in the Niger Delta. Chevron also has interests in deepwater projects, particularly its largest deepwater discovery Agbami.

Total and Eni are the fourth and fifth largest oil producers in the country, producing 179,000 bbl/d and 96,000 bb/d in 2011, respectively. Total operates several offshore projects and one onshore. Total is the operator of the Usan deepwater field that came online in July 2012. Total's smaller share of production has been unaffected in recent years whereas Eni/Agip has had some incidents, specifically at the Brass River terminal that have shut-in varying volumes of production since December of 2006.

## Exports

In 2011, Nigeria exported approximately 2.2-2.3 million bbl/d of crude oil, according to an analysis of data from the Global Trade Atlas (GTA), APEX Tanker Data (Lloyd's Maritime Intelligence Unit), and OPEC. Crude production estimates are sometimes less than crude export estimates for Nigeria due to oil theft, particularly from the wellhead, that reduces the amount of official oil sales and makes it difficult to estimate production.

Nigeria is an important oil supplier to the United States. In 2011, 767,000 bbl/d of crude (33 percent) of Nigeria's crude exports were sent to the United States, making Nigeria the fourth largest foreign oil supplier to the United States. Although Nigeria's high-quality light, sweet crude is a preferred gasoline feedstock, United States imports of Nigerian crude have decreased in volume and as a share of total imports in 2011, with the trend continuing in 2012. According to an EIA article published earlier this year, although total crude imports into the United States are falling, imports from Nigeria have declined at a steeper rate. The main reasons underlying this trend are that some Gulf Coast refiners have reduced Nigerian imports in favor of domestically-produced crude, and that two refineries in the U.S. East Coast, which were significant buyers of Nigerian crude, were idled in late 2011. As a result, Nigerian crude as a share of total United States imports has fallen to 5 percent in the first half of 2012, down from 10 and 11 percent in the first half of 2011 and 2010, respectively, according to EIA.

Despite shut-in production, Nigerian oil trade patterns appear to have remained stable over the past several years, most of which can be attributed to capacity additions and shifting world demand. Other major importers of Nigerian crude oil include Europe (28 percent), India (12 percent), Brazil (8 percent), Canada (5 percent), and South Africa (3 percent). According to the Energy Intelligence Group's International Crude Oil Market Handbook, Nigeria has about 20 exported crude streams and most are light, sweet grades, with gravities ranging from API 29 — 47 degrees and low sulfur contents of 0.05 — 0.3 percent.

## Downstream

#### Refining

In 2011, Nigeria consumed approximately 286,000 bbl/d of petroleum, according to EIA estimates, and about 180,000 bbl/d of which was gasoline, according to estimates from OPEC's Annual Statistical Bulletin. The country has four refineries (Port Harcourt I and II, Warri, and Kaduna) with a combined capacity of around 445,000 bbl/d, according to OGJ. As a result of poor maintenance, theft, and fire, none of these refineries have ever been fully operational. In 2009 and part of 2010 particularly low refinery runs forced the country to import about 85 percent of its fuel needs. In 2011, the operational capacity at refineries averaged 24 percent, slightly higher than the 22 percent in the previous year. Refineries have never reached full production capacity due to operational failures and sabotage, mainly on crude pipelines feeding refineries. Refinery utilization rates may improve in 2013 if the planned turn-around maintenance is performed.

For several years, the government has planned the construction of new refineries, but the lack of financing has caused several delays. As part of the PIB energy sector reforms, the government also plans to liberalize domestic fuel prices and privatize the refining sector. In the meantime, according to Business Monitor International, NNPC has signed contracts to swap crude for products under yearly contracts with Trafigura, an oil trading company, and Ivory Coast's national refiner SIR.

#### Oil infrastructure

Nigeria has over a dozen domestic crude oil pipelines that funnel crude to export terminals and domestic refineries. The pipelines run from 31 miles to as long as 383 miles, through mostly rural or swampy areas, making them difficult to police. Most of the pipeline systems are jointly owned by the major IOCs and NNPC, while the export terminals are operated by Shell (Forcados and Bonny terminals), ExxonMobil (Qua Iboe terminal), Chevron (Escravos and Pennington terminals) and Eni (Brass terminal). There are also several floating production, storage and offloading (FPSO) vessels that facilitate exports from deepwater offshore fields.

#### Domestic fuel prices & subsidies

On January 1, 2012, the Nigerian government removed the federal government fuel subsidy on the grounds that it caused market distortions, encumbered investment in the downstream sector, supported economic inequalities (as rich fuel-importing companies were the main beneficiaries), and created a nebulous channel for fraud. However, the government quickly reversed course about two weeks later and reinstated a partial subsidy as public outcry and massive strikes organized by oil and non-oil unions threatened to shut down oil production completely. Many Nigerians consider the fuel subsidy a key benefit of living in the oil-rich country.

Prior to the subsidy removal, the pump price of fuel was 65 naira ($0.40) per liter compared to the actual cost of around 139 naira per liter. According to the United Nations, the fuel subsidy costs the Nigerian government annually 1,200 billion naira ($7.6 billion), or 2.6 percent of the country's GDP. Subsequent to the removal, the government restored a partial subsidy, requiring consumers at the pump to pay 97 naira per liter ($0.60), as opposed to the new price of 141 naira per liter. Controversy over Nigeria's fuel subsidy resurfaced in August 2012, as Nigerian oil marketing associations launched an open-ended strike over unpaid subsidies. The associations accused the government of stopping payments around March/April, though the government denies this claim. Tensions between Nigerian fuel importers and the government have been high since the government launched an investigation of the industry to mitigate subsidy mismanagement. The investigation has led to the arrest of about a dozen marketers and the suspension of seven companies that were accused of siphoning funds and inflating prices, according to IHS Cera. Debate over Nigeria's fuel subsidy will continue among government officials, oil marketing associations, unions, and citizens, especially since the most recent version of the PIB attempts to deregulate the downstream sector. However, it is unclear how the PIB will affect fuel subsidies. Meanwhile, under the 2012 budget, the government is expected to subsidize at least 104,000 bbl/d of fuel imports for the remainder of 2012, which falls short by about 160,000-170,000 bbl/d of what is needed. According to PFC Energy, the government overestimated fuel subsidy savings and underestimated subsidy arrears' claims in 2012, and may have to access the Excess Crude Account to avoid strong public discontent. Map: Niger Delta Oil Infrastructure Source: U.S. government # Natural gas  Nigerian LNG exports to the U.S. substantially declined in 2011, while the country's LNG exports to Japan more than tripled in 2011. Nigeria had an estimated 180 trillion cubic feet (Tcf) of proven natural gas reserves as of the end of 2011, according to the OGJ, making Nigeria the ninth largest natural gas reserve holder in the world and the largest in Africa. Despite holding a top 10 position for proven natural gas reserves, Nigeria produced about 1 Tcf of dry natural gas in 2011 and ranked as the world's 25th largest natural gas producer. The majority of the natural gas reserves are located in the Niger Delta and, therefore, the sector is also impacted by the same security and regulatory issues affecting the oil industry. Most of Nigeria's marketed natural gas is exported as Liquefied Natural Gas (LNG), with the remainder consumed domestically and other portions exported regionally via the West African Gas Pipeline. Shell Nigeria Gas Limited (SNG), a Shell-owned gas sales and distribution company, also delivers Compressed Natural Gas (CNG) to industries as far as 62 miles away from existing pipelines. Dry natural gas production grew for most of the last decade until Shell declared a force majeure on gas supplies to the Soku gas-gathering and condensate plant in November 2008. Shell shut down the plant to repair damages to a pipeline connected to the Soku plant that was sabotaged by local groups siphoning condensate. The plant reopened nearly 5 months later, but was shut down again for most of 2009 for operational reasons. The Soku plant provides nearly half of the feed gas to Nigeria's sole LNG facility; therefore, its closure led to a reduction in Nigeria's natural gas production, particularly from Shell's fields in the Niger Delta, and a 33 percent decline in LNG exports in 2009. Gas production partially recovered in 2010 after the plant reopened. ## Sector organization For the most part, the same national regulatory bodies and international oil companies (IOCs) involved in Nigeria's oil industry are also the actors involved in the gas industry. The Nigerian Gas Company Limited (NGC), a subsidiary of NNPC, is tasked with the marketing, transmission, and distribution of gas and oversees pipeline projects. The PIB proposes to divide the NGC into two organizations: the midstream National Gas Transportation Company, and a downstream gas marketing company. Like in the oil industry, NNPC holds interest in gas projects alongside international oil companies. #### International oil companies Shell dominates gas production in the country, as the Niger Delta, which contains most of Nigeria's gas resources, also houses most of Shell's hydrocarbon assets. Shell produced 707 MMcf/d of gas in 2011 and its latest gas project, the Gbaran-Ubie integrated oil and gas project, achieved peak gas production of 1 Bcf/d in early 2011. Gbaran-Ubie's gas is delivered to domestic power plants and to NLNG for export. The second largest gas producer, Total, produced 534 MMcf/d in 2011. Total, along with Eni, is developing the Brass LNG facility, which will comprise two trains with the capacity of processing 5 million metric tons per year of LNG sometime after 2014. Eni was the third largest natural gas producer in Nigeria in 2011. The company's natural gas output grew by almost 40 percent in the last three years and reached 354 MMcf/d in 2011. Chevron produced 343 MMcf/d in 2011 and is majority owner of major gas projects in the country, such as the WAGP and the Escravos GTL plant, as noted above. ## Gas flaring Since Nigeria's oil fields lack the infrastructure to produce and market associated natural gas, much of it is flared. According to the National Oceanic and Atmospheric Administration (NOAA), Nigeria flared 536 Bcf of natural gas in 2010 – or about a third of gross natural gas produced in 2010, according to NNPC. In 2011, the NNPC claimed that flaring cost Nigeria US$2.5 billion per year in lost revenue.

The Nigerian government has been working to end natural gas flaring for several years, but the deadline to implement the policies and fine oil companies has been repeatedly postponed with the most recent deadline being December 2012, which appears unlikely to be enforced. In 2009, the Nigerian government developed a Gas Master Plan that promotes investment in pipeline infrastructure and new gas-fired power plants to help reduce gas flaring and provide much-needed electricity generation. However, progress is still limited as security risks in the Niger Delta have made it difficult for IOCs to construct infrastructure that would support gas monetization.

## Gas to liquids (GTL)

A Chevron-operated Escravos Gas to Liquids (GTL) project is currently underway. The project is a joint venture with NNPC and South Africa's Sasol and began in 2008. Escravos GTL has faced multiple delays and cost overruns, but is currently scheduled to be operational by 2013. The project will convert 325 million cubic feet of natural gas per day into 33,000 barrels of liquids, principally synthetic diesel, to supply clean-burning, low-sulfur diesel fuel for cars and trucks, according to Chevron.

## Exports

#### Liquefied natural gas (LNG)

A significant portion of Nigeria's marketed natural gas is processed into LNG. In 2010, Nigeria exported 17.97 million metric tons (875 Bcf) of LNG, making Nigeria the fifth largest LNG exporter in the world and the largest LNG exporter in the Atlantic Basin. Furthermore, Nigeria's LNG accounted for 8 percent of the total supplied to the world market and 30 percent of LNG coming from the Atlantic Basin in 2010. However, although Nigeria's market share of LNG trade in the Atlantic Basin has been increasing, mainly due to decreased LNG exports from Algeria, the country's market share in the world has decreased from the 10 percent it once held to 7 percent, as reported by Nigeria Liquefied Natural Gas (NLNG) Limited in 2012. Nigerian LNG exports rose to 18.86 million metric tons (918 Bcf) in 2011, but due to no recent capacity increases and rising production from Qatar and Australia, Nigeria's world market share of LNG is slipping. Nigeria's LNG production capacity is currently 22 million metric tons per year, and any major increase is not expected to come online before 2015.

In 2010, most of Nigeria's LNG was exported to Europe (67 percent), mainly Spain (31 percent), France (16 percent) and Portugal (12 percent), with smaller amounts to Turkey, United Kingdom, and Belgium. Other export destinations include Asia (15 percent) and North America (14 percent). The U.S. imported 0.86 million metric tons (42 Bcf) of Nigerian LNG in 2010, providing 1 percent of total U.S. LNG imports.

In 2011, U.S. imports of Nigerian LNG significantly decreased to 0.05 million metric tons (2.5 Bcf), according to EIA data, which is the lowest level recorded since Nigerian LNG exports began. In 2011, more of Nigeria's LNG exports were sent to Japan and other Asian countries due to higher demand for LNG imports in those countries. Most notably, Nigerian exports to Japan more than tripled in 2011, as Japan's LNG demand increased due to the Fukushima nuclear accident.

The Nigeria Liquefied Natural Gas (NLNG) facility on Bonny Island is Nigeria's only LNG complex. NLNG partners, including NNPC (49 percent), Shell (25.6 percent), Total (15 percent), and Eni (10.4 percent), completed the first phase of the facility in September 1999. NLNG currently has six trains and a production capacity of 22 million metric tons per year (1.1 Tcf). A seventh train is under construction to increase the facility's capacity by 8 million metric tons per year. However, regulatory and political issues, particularly regarding the long-delayed PIB, have delayed the project's start date to beyond 2014.

Three additional LNG plants with a total of seven trains were expected to come online after 2012, but their expected start dates have been postponed beyond 2016. Plans include OK LNG (4 trains), Brass LNG (2 trains), and Progress LNG (1 train). These are in varying stages of development, and investment decisions will depend heavily on security, world LNG markets, and the final outcome of the PIB. Availability of natural gas for export will also depend on Nigerian efforts to expand the use of natural gas for domestic electricity generation – efforts that are included in both the Gas Master Plan and the PIB.

#### International pipelines

Nigeria began exporting some of its natural gas via the West African Gas Pipeline (WAGP) in 2011. The pipeline is operated by the West African Gas Pipeline Company limited (WAPCo), which is owned by Chevron West African Gas Pipeline Limited (36.7%), Nigerian National Petroleum Corporation (25%), Shell Overseas Holdings Limited (18%), Takoradi Power Company Limited (16.3%), Societe Togolaise de Gaz (2%), and Societe BenGaz S.A. (2%).

The 420-mile pipeline carries natural gas from Nigeria's Escravos region to Togo, Benin, and Ghana. WAGP links into the existing Escravos-Lagos pipeline and moves offshore at an average water depth of 35 meters. According to WAPCo, roughly 85 percent of the gas is used for power generation and the remainder for industrial applications. Current recipients are Volta River Authority's Takoradi Thermal Power Plant in Ghana and Electricity Community of Benin (CEB), a company co-owned by Benin and Togo. Exports should eventually reach initial capacity of 170 million cubic feet per day (MMcf/d) and plans are underway to expand capacity to as much as 460 MMcf/d and possibly extend the pipeline further west to Cote d'Ivoire.

As of early October 2012, the pipeline is shutdown due to a loss of pressure around the Lome segment that it experienced at the end of August 2012. The WAPCo has noted that maintenance to the damaged pipeline is planned for completion at the end of December.

Nigeria and Algeria continue to discuss the possibility of constructing the Trans-Saharan Gas Pipeline (TSGP). The 2,500-mile pipeline would carry natural gas from oil fields in Nigeria's Delta region to Algeria's Beni Saf export terminal on the Mediterranean and is designed to supply gas to Europe. In 2009, the NNPC signed a memorandum of understanding (MoU) with Sonatrach, the Algerian national oil company, to proceed with plans to develop the pipeline. Several national and international companies have shown interest in the project, including Total and Gazprom. Security concerns along the entire pipeline route, increasing costs, and ongoing regulatory and political uncertainty in Nigeria have continued to delay this project.

# Electricity

Nigeria's electricity sector is relatively small. Brazil and Pakistan, two countries with similar population sizes, generate 24 times and 5 times more power than Nigeria, respectively. Bangladesh, a country slightly smaller in population and with a smaller gross domestic product (GDP) than Nigeria, produces nearly twice as much electricity as Nigeria. The latest EIA estimates show that Nigeria's net generation was 18.8 billion kilowatthours (KWh) in 2009. Installed electricity capacity has remained relatively stable over the last decade at 5.9 GW, although net generation has slightly decreased from its peak of 23 billion KWh in 2004, mainly due to a decline in hydroelectric power.

The majority of electricity generation comes from thermal power plants (77 percent), with about two-thirds of thermal power derived from natural gas and the rest from oil. Hydroelectricity (23 percent), the only other source of power generation, has decreased gradually from its peak of 8.2 billion KWh in 2002 to 4.5 billion KWh in 2009. Nigeria's electricity net consumption was 17.7 billion kWh in 2009, slightly less than generation, and exported most of the remainder to Niger through an agreement under the West African Power Pool.

According to a World Bank report, Nigeria experienced power outages on average for 46 days per year from 2007-2008, and outages lasted almost 6 hours on average. Population growth coupled with underinvestment in the electricity sector has led to increased power demand without any significant increases in capacity, in addition to inadequate maintenance, insufficient feedstock and an inadequate transmission network. Businesses often purchase costly generators to use as back-up during outages and the majority of Nigerians use traditional biomass, such as wood, charcoal, and waste, to fulfill household energy needs, such as cooking and heating.

Nigeria's electricity sector is divided into three sub-sectors: existing Federal Government of Nigeria (FGN) Power Generation facilities, Independent Power Projects (IPPs), and National Integrated Power Projects. The majority of power stations, both thermal and hydro, are FGN facilities funded by the government, while IPPs are backed by the private sector. The largest IPP and power plant in Nigeria is the 650 megawatt (MW) Afam VI Power Generating Plant owned by Shell. According to Shell, between 14-24 percent of overall generation contributes to the national grid.

The National Integrated Power Project (NIPP) is a plan launched by the Nigerian government to construct multiple new power plants and reduce gas flaring for feedstock. However, plans to bring NIPPs online have repeatedly been delayed, as government plans to privatize electricity generation and distribution companies have been slow. Two major challenges of privatization is unbundling the state-owned Power Holding Company of Nigeria, which was established to regulate pricing and competition, and publically managing the expected 88 percent rise in electricity tariffs once privatization is underway. The Nigeria Electricity Regulatory Commission has also said that new tariffs will be imposed on selected cities.

Currently, there are some IPPs under construction such as ExxonMobil's 388-MW gas-fired plant in Bonny and ABB's 450-MW gas and steam turbine in Abuja. Nonetheless, according to the government, the country needs \$10 billion of investment a year for at least a decade to meet its power sector needs. In addition to funds, Nigeria's ability to meet power demand heavily relies on its ability to reduce gas flaring, increase gas distribution infrastructure, and diversify power generation sources.

# Sources

• Afroil: Africa Oil and Gas Monitor (Newsbase)
• BBC News
• BP Statistical Review of World Energy, 2011
• Business Monitor International
• CIA World Factbook
• Chevron
• Economist
• Energy Intelligence Group
• Eni
• Eurasia Group
• ExxonMobil
• FACTS Global Energy
• Financial Times
• Global Trade Atlas
• IHS Global Insight
• International Energy Agency (IEA)
• International Maritime Organization
• International Monetary Fund
• Lloyd's APEX Database (Analysis of Petroleum Exports)
• New York Times
• Oil and Gas Journal
• OPEC Annual Statistical Bulletin
• Petroleum Africa
• Petroleum Economist
• Petroleum Intelligence Weekly
• Reuters
• Rigzone
• Shell
• Total
• World Bank
 Disclaimer: This article is taken wholly from, or contains information that was originally published by, the Energy Information Administration. Topic editors and authors for the Encyclopedia of Earth may have edited its content or added new information. The use of information from the Energy Information Administration should not be construed as support for or endorsement by that organization for any new information added by EoE personnel, or for any editing of the original content.
Glossary

### Citation

Administration, E. (2013). Energy profile of Nigeria. Retrieved from http://www.eoearth.org/view/article/152513

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text:
Question

# A square and a rectangle field with measurement as given in the figure have the same perimeter. Which field has a larger area?

Hint: First find the other side of the rectangle by equating the perimeter of square and rectangle. Now use formula for area of square and rectangle to get the final answer.

It is given that perimeter of rectangle = perimeter of square.. (1)
Step 1: Let us suppose that side of square = s
Perimeter of square = $60 + 60 + 60 + 60 = 240$= (4s)
From (I)
$2\left( {l + b} \right) = 240 \\ \Rightarrow 2\left( {80 + b} \right) = 240 \\ \Rightarrow 160 + 2b = 240 \\ \Rightarrow 2b = 240 - 160 \\ \Rightarrow 2b = 80 \\ \Rightarrow b = 40 \\ \\$
So, $l = 80,{\text{ b = 40}}$
Step 2: Now we have to find out whether the area of square has the larger area or the area of rectangle has the larger area:
We know that:
Area of square=$side \times side$
=$60 \times 60 \\ = 3600 \\$
Area of rectangle= $l \times b$
$= 80 \times 40 \\ = 3200 \\$
Therefore, the area of the square is greater than the area of the rectangle that is 3600.

Note: Whenever we face such types of questions, the key concept is that we have to first calculate the perimeter of the square then the perimeter of the rectangle. First, we calculate the perimeter of the square like we did in the question. With the help of the perimeter of the square we will calculate the breadth of the rectangle by applying the formula$2\left( {l + b} \right)$. Second step is to calculate the area of the rectangle and area of square and to check whether the fig a has the larger area or the fig b has the larger area. After applying the formula we get our required answer , like in the above question the area of the square is greater than the area of the rectangle.