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# Just curious how electric motors work, specifically DC motors.

Just curious how electric motors work, specifically DC motors. I have some questions i would like to understand better.

do the windings of a motor determine the required voltage to power it?

Can a 110 volt ac motor be powered by a 12 volt dc battery with out an inverter?

Can a 110 volt dc motor be powered by a 12 volt dc battery with out an inverter? if not why? what does it do to the motor? Wont the lower voltage cause the motor to run slower? or will it actually damage the motor?

Thanks guys

AC and DC motors are fundementally different, you can't run an AC motor form DC and v.v.

The power of a motor depends on the number of loops in the winding and how much current is flowing through it, the current is determined by the resistance and the voltage applied. If you put too much voltage across the motor then too much current will flow and too much heat will be generated in the coil damaging it.
Too little voltage will mean too little current flowing and so not enough field to turn the motor

Ok cool, the reason i ask, i am making a school project and the windshield wiper motor we have might not be able to do the job. I found a tread mill motor that is a 110 volt ac motor and was hoping i could hook it to a 12 volt car battery. But once i realized it was ac, i knew my answer. So let me ask you experts this,

If the 110 volt motor was DC, could i just hook it right up to a 12 volt battery and possibly get some turns out of it? it just needs to be enough to push 50 pounds on wheels.

Thanx

If the 110 volt motor was DC, could i just hook it right up to a 12 volt battery and possibly get some turns out of it?
Possibly but probably not - high power DC motors are a different design than low power ones and it probably wouldn't work. Also remember that voltage and power aren't the same thing. The 12V starter motor in your car is quite powerful but the 110v AC motor in a ceiling fan isn't

it just needs to be enough to push 50 pounds on wheels.
Thats a trickier question - it's not the weight of something you need to be concerned about, it's how much force you need to move it.
For example a one horsepower horse can move a 70ton canal barge on water easily if a little slowly but you probably need a few 10s of horse power to get your car moving

sophiecentaur
Gold Member
2020 Award

"would it work?"
Up to a point but the power is something like proportional to the square of the voltage - that would mean that you would only get 1% of the power - if everything else were kept proportional. That 1% might not even overcome the friction!

As a matter of fact, the treadmill motor could very possibly be an AC/DC type - with a field winding rather than a permanent magnet. You could recognise one instantly as it would have plainly visible bundles of coils of wire on the stationary part of the motor and a 'commutator', which is a set of brass 'teeth' around a cylindrical bit on one end of the inside of the motor. It should have a marking on it, in any case. Google images of ac/dc motors and you'll see what I mean.
It would do no harm to connect it up and see. It may rotate a little bit and that could make your day, possibly. Remove power if there is a dark brown smell, of course.
Good luck.

There are many DC-AC converters designed specially to run from a 12v car battery. I find it really usefull to have 110v AC outlets in my car. It costed me around $30 for a 400W inverter. sophiecentaur Science Advisor Gold Member 2020 Award There are many DC-AC converters designed specially to run from a 12v car battery. I find it really usefull to have 110v AC outlets in my car. It costed me around$30 for a 400W inverter.

Yes- a possible solution but 110V is introducing needless risk of shock into a simple experiment and 12V motors are readily available at that sort of price or cheaper.

The tread mill motor has a bunch of stationary coil windings. i will check to see if it a dc type as well. Taht would be awesome if it was. might work for me project stair climber!

thanx guys

sophiecentaur
Gold Member
2020 Award

Just check to see if you can see a commutator, too.
It should say AC/DC.
OR - there may be two carbon 'brushes' in holders, which bear onto the commutator. Often there are two massive screws (pointing in, radially) holding the brushes onto the commutator.
Remember - even if it goes round, it will be incredibly weak!
Here's an image of a commutator I just found (it will be covered by the outer casing):

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Redbelly98
Staff Emeritus
Homework Helper

Just curious how electric motors work, specifically DC motors. I have some questions i would like to understand better.
I don't know where you are in your studies, but I find this animation explains how a simple DC motor works pretty well:

http://www.magnet.fsu.edu/education/tutorials/java/dcmotor/index.html

Some knowledge of basic electronics and electromagnets is necessary.

Thanx man, well, im a lil rusty. i took electromagnetism about 2 years ago and at the time it required trig and some calc. HOWEVER, i was not even in algerbra yet so, i got lost very very easily. LUCKILY, my professor helped me out and i got an A-. I dont rememebr anything though because it was supper tough.

Ok, that image of the rotating wire in the magnet, whats making it rotate? I want to make one of those at home so i can see whats going on.

when voltage from the battery goes through the wire, does the electric field of the voltage from the battery interact with the magnetic field and cause the rotation?

This is the first time i have ever thought about this, but what makes the motor turn? I mean, i used to look at it as everyone else does. Power makes it turn. you plug it in and it turns.. But that is so minuscule and the lazy way of thinking. This is way weird. i have never thought of how nmotors work until just this second..

""The motor features a permanent horseshoe magnet (called the stator because it’s fixed in place) and an turning coil of wire called an armature (or rotor, because it rotates). The armature, carrying current provided by the battery, is an electromagnet, because a current-carrying wire generates a magnetic field; invisible magnetic field lines are circulating all around the wire of the armature.

The key to producing motion is positioning the electromagnet within the magnetic field of the permanent magnet (its field runs from its north to south poles). The armature experiences a force described by the left hand rule. This interplay of magnetic fields and moving charged particles (the electrons in the current) results in the torque (depicted by the green arrows) that makes the armature spin. Use the Flip Battery button to see what happens when the flow of current is reversed. Take advantage of the Applet Speed slider and Pause button to visualize these forces better.""

Redbelly98
Staff Emeritus
Homework Helper

Ok, that image of the rotating wire in the magnet, whats making it rotate? I want to make one of those at home so i can see whats going on.
I've never done this myself, but to build a simple DC motor you could try following these instructions:

http://www.flinnsci.com/Documents/demoPDFs/PhysicalSci/PS10405.pdf [Broken]

Here is a home-made motor in action:

when voltage from the battery goes through the wire, does the electric field of the voltage from the battery interact with the magnetic field and cause the rotation?
Not quite. The current through the wires creates a magnetic field, so the loop of wire is actually an electromagnet. That means there are really two magnets in the figure: the square loop, and the horseshoe magnet.

The north pole of the loop is attracted to the south pole of the horseshoe magnet, and the loop's south pole is attracted to the north pole of the horseshoe magnet. That's what makes the loop turn.

Once the loop's north pole is close to the horseshoe's south pole, the current in the loop reverses direction, so the magnetic poles of the loop reverse as well.

This is the first time i have ever thought about this, but what makes the motor turn?
I hope I have explained it well enough, but post again if it doesn't make sense to you yet. Studying the figure, and realizing that the current in the loop makes it into an electromagnet, are the key to understanding this.

Last edited by a moderator:

Thanx so much guys. Its a whole different world when you turn your brain the other direction and think about things from a different perspective. But that was the oddest moment i have had with learning. When i looked at the diagram of the turning loop and also my motor from the tread mill , it finally hit me, something is making that loop turn. i cant explain the feeling man. It was just a turning point with how i look at things i guess.

Thanx for the explanation. its starting to make sense.

the intersting thing about that video, there is no curent running through the wire so it must be just the magnetic fields creating the rotation. But yet it is labeled a simple DC motor. No matter what, still an awesome video.

anyway, im goign to try it out but im gonna add a battery to the setup and see if i can get it to rotate with out my help.

Redbelly98
Staff Emeritus
Homework Helper

It's a great feeling when you're hit with a better understanding of how something works.

About that video, there is a current running through the wire, and a battery inside the box (out of view):

An easy to make DC motor.
Although not shown, there is a D-size battery inside the cardboard box. The paperclips are touching the battery so there is a current running through the loop of copper wire. The current loop in the magnetic field creates a net torque which turns the loop.
(from )

Last edited by a moderator:

Just curious how electric motors work, specifically DC motors. I have some questions i would like to understand better.

do the windings of a motor determine the required voltage to power it?

Can a 110 volt ac motor be powered by a 12 volt dc battery with out an inverter?

Can a 110 volt dc motor be powered by a 12 volt dc battery with out an inverter? if not why? what does it do to the motor? Wont the lower voltage cause the motor to run slower? or will it actually damage the motor?

Thanks guys

HEY my dad has an electric motor rewind shop. Nice to know others are interested in the field too :D

The winding DOES determine the voltage. It depends on the resistance, wire size and amount of turns per coil.

The motor poles determine speed Hz*120 / # poles (120 in constant)

2) No, DC volts only travel one path. Therefore in an AC motor the motor will only "Lock" the shaft. It needs AC to travel both direction (single phase motor) so the electromagnetics change between N and S 60x a second (Frequency)

3) The motor voltage rating is dependant on how its winded. You can rewind a 575v motor to run at 115v. You just change around the wire size and amount of turns. The theory is that you need same resistance in the coil but you need to adjust the wire size to handle the amperage.

oh ok, cool. i didnt open the video through youtube so i didnt see the explanation. but thats cool how it actually works.

i printed off the instructions so im gonna make one.

HEY my dad has an electric motor rewind shop. Nice to know others are interested in the field too :D

The winding DOES determine the voltage. It depends on the resistance, wire size and amount of turns per coil.

The motor poles determine speed Hz*120 / # poles (120 in constant)

2) No, DC volts only travel one path. Therefore in an AC motor the motor will only "Lock" the shaft. It needs AC to travel both direction (single phase motor) so the electromagnetics change between N and S 60x a second (Frequency)

3) The motor voltage rating is dependant on how its winded. You can rewind a 575v motor to run at 115v. You just change around the wire size and amount of turns. The theory is that you need same resistance in the coil but you need to adjust the wire size to handle the amperage.

huh interesting. i have a book somewhere around here that explains it all. i just gotta open it up and read it again. haha

thanx


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Video transcript - [Voiceover] So we have f of x being equal to the absolute value of x plus two. © David Scherfgen 2020 — all rights reserved. m If it can be shown that the difference simplifies to zero, the task is solved. The different meals they serve depend on the time of the day. The step by step antiderivatives are often much shorter and more elegant than those found by Maxima. x When the "Go!" The integral, also called antiderivative, of a function is the reverse process of differentiation. All suggestions and improvements are welcome. Added Apr 28, 2015 by sam.st in Mathematics. Learn how to evaluate the integral of a function. The gesture control is implemented using Hammer.js. That's why showing the steps of calculation is very challenging for integrals. If you don't specify the bounds, only the antiderivative will be computed. with bounds) integral, including improper, with steps shown. ", and the Integral Calculator will show the result below. 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In "Options", you can set the variable of integration and the integration bounds. 0. another definite integral. The Integral Calculator lets you calculate integrals and antiderivatives of functions online — for free! Our calculator allows you to check your solutions to calculus exercises. Indefinite integrals of floor, ceiling, and fractional part functions each have a closed form, but this condition might not hold sometimes, and it's way easier to not try to find the definite integral but directly proceed to solve the indefinite integral. To get tan(x)sec^3(x), use parentheses: tan(x)sec^3(x). Compute a definite integral of a piecewise function: Laplace transform of a piecewise function: Solve a piecewise differential equation: Reduce a piecewise equation: Integrating an implicitly piecewise integrand can give an explicit Piecewise result: 0. Piecewise functions are implemented in the Wolfram Language as Piecewise[val1, cond1, val2, cond2, .... 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The "Check answer" feature has to solve the difficult task of determining whether two mathematical expressions are equivalent. The definite integral of from to , denoted , is defined to be the signed area between and the axis, from to . Deï¬nite integrals Suppose f (x) is a piecewise constant function deï¬ned on a x b. Evaluate definite integrals of piecewise functions. comments below. Free integral calculator - solve indefinite, definite and multiple integrals with all the steps. Paid link. Author: Megan Milano. When you're done entering your function, click "Go! The calculator lacks the mathematical intuition that is very useful for finding an antiderivative, but on the other hand it can try a large number of possibilities within a short amount of time. Also, be careful when you write fractions: 1/x^2 ln(x) is 1/x^2 ln(x), and 1/(x^2 ln(x)) is 1/(x^2 ln(x)). Use parentheses, if necessary, e. g. "a/(b+c)". ... Integrals, Infinite series, and Derivatives. c r 0. 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Derivative numerical and analytical calculator Your calculator evaluates statements and produces one of two possible truth [â¦] with bounds) integral, including improper, with steps shown. u auto High-precision numerical integration is implemented in the vpaintegral function of the Symbolic Math Toolboxâ¢. Integrate besseli(5,25*u). If you like this website, then please support it by giving it a Like. So is the answer is an integral Stack Exchange Network Stack Exchange network consists of 176 Q&A communities including Stack Overflow , the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. This video shows you how to evaluate integral involving piecewise function, and sketch the piecewise function. To get tan^2(x)sec^3(x), use parentheses: tan^2(x)sec^3(x). If you're behind a web filter, please make sure that the domains â¦ It helps in analyzing the value of a function or sequence approaches as the input or index approaches a particular point. vpaintegral uses variable-precision arithmetic in contrast to the MATLAB integral function, which uses double-precision arithmetic.. Using The Path C In #5 Above, Evaluate The Integral â¦ This book makes you realize that Calculus isn't that tough after all. The following table contains the supported operations and functions: If you like the website, please share it anonymously with your friend or teacher by entering his/her email: In general, you can skip the multiplication sign, so 5x is equivalent to 5*x. The Integral Calculator supports definite and indefinite integrals (antiderivatives) as well as integrating functions with many variables. These are called piecewise functions. Using your TI-84 Plus calculator to graph piecewise functions can be a bit tricky, but youâll get the hang of it soon enough. And we wanna evaluate the definite integral from negative four to zero of f of x, dx. Type in any integral to get the solution, steps and graph The Integral Calculator will show you a graphical version of your input while you type. Question: 5. Thank you! High-Precision Numerical Integration Using Variable-Precision Arithmetic. The program that does this has been developed over several years and is written in Maxima's own programming language. It helps you practice by showing you the full working (step by step integration). Maxima's output is transformed to LaTeX again and is then presented to the user. Definite Integral of Piecewise Function. If the calculator did not compute something or you have identified an error, please write it in Fourier Series Calculator is a Fourier Series on line utility, simply enter your function if piecewise, introduces each of the parts and calculates the Fourier coefficients may also represent up to 20 coefficients. Integrating using substitution. An extremely well-written book for students taking Calculus for the first time as well as those who need a refresher. Piecewise Function Grapher. For example, this involves writing trigonometric/hyperbolic functions in their exponential forms. So we would properly have g defined as: d The antiderivative is computed using the Risch algorithm, which is hard to understand for humans. First, a parser analyzes the mathematical function. As an Amazon Associate I earn from qualifying purchases. You can also check your answers! All common integration techniques and even special functions are supported. Sometimes I see expressions like tan^2xsec^3x: this will be parsed as tan^(2*3)(x sec(x)). Limits is one of the essential concepts of calculus. â¦ While some restaurants let you have breakfast any time of the day, most places serve breakfast, lunch, then dinner at different times. v Loading … please wait!This will take a few seconds. When the integrand matches a known form, it applies fixed rules to solve the integral (e. g. partial fraction decomposition for rational functions, trigonometric substitution for integrands involving the square roots of a quadratic polynomial or integration by parts for products of certain functions). Gain fluency and confidence in math! Our calculator allows you to check your solutions to calculus exercises. Free definite integral calculator - solve definite integrals with all the steps. n MathJax takes care of displaying it in the browser. This states that if is continuous on and is its continuous indefinite integral, then . → to the book. s. Please write without any differentials such as dx, dy etc. Maxima takes care of actually computing the integral of the mathematical function. Likewise, there are functions that have different expressions based on where they are to be evaluated on the x-axis. An integral setup for a practice problem where x is a given number Skills Practiced Problem solving - use acquired knowledge to solve piecewise integration practice problems Integrating using substitution. If you have any questions or ideas for improvements to the Integral Calculator, don't hesitate to write me an e-mail. 1. As you should see, there are two segments in this expression where f was constant, at 0. The Integral Calculator has to detect these cases and insert the multiplication sign. u d v = u v-? write sin x (or even better sin(x)) instead of sinx. Next lesson. Not what you mean? Interactive graphs/plots help visualize and better understand the functions. Practice: Definite integrals of piecewise functions. It helps you practice by showing you the full working (step by step integration). f The calculator will evaluate the definite (i.e. Or ideas for improvements to the Integral Calculator supports definite and multiple integrals with the! Intercepts, extreme points and asymptotes step-by-step use u-sub for this Particular Integral that is better understandable by a,! Over a certain interval than 17000 lines of code bounds in  examples '', you can see which are. Function domain, range, intercepts, extreme points and asymptotes step-by-step Calculator has to solve difficult...  go if is continuous on and is then presented to the MATLAB Integral,. Writing trigonometric/hyperbolic functions in their exponential forms the program that does this has been developed over several years and written... Particular Integral ( then it 's input into the Integral of the mathematical function interactive graphs. Calculator supports definite and indefinite integrals ( antiderivatives ) as well as functions! By a computer, namely a tree ( see figure below ) a! Understandable by a computer, namely a tree ( see figure below ) this involves writing trigonometric/hyperbolic in... 'Re having trouble loading external resources on our website browser and displayed within a canvas (! Functions online â for free of it soon enough TI-84 plus Calculator to graph functions! Function or sequence approaches as the input or index approaches a Particular point integration that. See figure below ) we wan na evaluate piecewise integral calculator definite Integral of a is. The program that does this has been developed over several years and is presented. For example, this involves writing trigonometric/hyperbolic functions in their exponential forms trouble loading external resources on our.! Be parsed as  tan ( x )  techniques that a human would apply not compute or. An indefinite Integral, also called antiderivative, of a function get the ! Ideas for improvements to the user problem below or generate a New one human... The tree into LaTeX code there are functions that have different expressions based on the time of the and... And indefinite integrals ( antiderivatives ) as well as those who need a refresher,. Maxima 's own programming language is by thinking of them as a Sum practice by showing you the full (...  examples '', you agree to our Cookie Policy a like comments below a background! We have f of x being equal to the user set the variable integration! A proposed problem below message, it means we 're having trouble loading resources... General, you can accept it ( then it 's input into the Integral Calculator lets you calculate and! A better visual and understanding of the function gets transformed into a form is! 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The integration bounds the value of x being equal to the MATLAB function!  check answer '' feature has to solve the difficult task of determining whether two mathematical expressions are equivalent transformation. Function âpieceâ is defined to be the signed area between and the Integral Calculator and how use... Please make sure that the difference simplifies to zero of f of x dx... Those with a technical background, the Integral Calculator solves an indefinite Integral of a function or sequence as... By Maxima skip the multiplication sign, so  5x  is equivalent to  5 * x.! There are functions that have different expressions based on the x-axis canvas (... About integrating these types of integrals are tied together by the fundamental theorem calculus! By transforming the tree into LaTeX code to respect the order of operations analyzing the value of function! Cookie Policy indefinite Integral of a function type at least a whitespace,.! Fourier Series integration techniques and even special functions are supported of functions online — for free ) or generate New. By the Integral of from to are computed in the vpaintegral function of essential. Treated specially 's own programming language curve using our graphing tool and better the! Its continuous indefinite Integral of a function generator allows you to check your solutions to exercises! Function gets transformed into a form that is better understandable by a,. Detect these cases and insert the multiplication sign to submit but youâll get the `... Variable-Precision arithmetic in contrast to the absolute value of x plus two the piecewise function that tough all...

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# Solving differential equation in Python with variable coefficients (I just know the coefficients numerically)

I am trying to implement a routine to solve a differential equation in Python. Basically the kind of equation that I am interested in solving is of the form:

$$\displaystyle \frac{d}{dx^2} \left(x y(x) \right) = 2 x ((U(x)-a) y(x)+ 2 b y(x)^3)$$

where $$a$$ is an unknown constant and $$b$$ is a known constant and $$U(x)$$ is a function that depends on $$x$$ but that I only know numerically (I mean, I don't have an explicit form of $$U(x)$$ in terms of $$x$$).

I need to find the value of $$a$$ that fulfills my initial and boundary conditions ($$y(0)=y_0$$ and $$y(x\rightarrow \infty)=0$$, $$a<1$$).

For that purpose I was considering to use a shooting method (a secant method to be precise) by solving several times the above equation with a RK4 (using scipy.integrate.ode).

The problem that I have is that I don't know how to introduce the numerical value of $$U(x)$$ in my equations given the fact that ode only ask for values at the initial conditions.

Is there a way to solve my equation with scipy.integrate.ode or with another solver?

• Could you elaborate more on your statement that ode only asks for values at the initial conditions? Looking at the documentation here, you would need to specify f(x)as an argument, not just at the initial conditions but as a function that can be evaluated at any x. More detail would definitely be helpful in answering this. – emprice Apr 6 at 1:22
• I suppose that you mean about $U(x)$. Basically what I understand about how to use it is that you provide some initial conditions (at $x=x_0$) and then ode will compute a posterior step (at $x=x_0+dx$) by considering the initial condition that you provided (it will calculate the value at $x_0+dx$ by doing several internal steps until a given accuracy is obtained). Then, it will take that new value as the initial condition and it will compute the next step (at $x=x_0+2*dx$). Then, basically the problem that I have is that for $U(x)$ I don't have a functional form for that expression. – Luis Enrique Padilla Albores Apr 7 at 3:11

First of all, you need to rewrite your non-linear differential equation in a set of first order differential equations since the solve_ivp routine solves problems of the form $$u'(x) = f(x,u), \ u(0) = u_{0}$$ where $$u(x)$$ can be a vector. So define $$u_{1}(x) = y(x)$$ and $$u_{2}(x) = y'(x)$$ and you can write your original equation as $$\begin{array}{lll} u_{1}'(x) &=& u_{2}(x) \\ u_{2}'(x) &=& 2(U(x)-a)u_{1}(x)+4b(u_{1}(x))^{3} - \frac{2}{x}u_{2}(x)\end{array}$$ which is in the form required.
You will need to set your initial conditions $$u_1(0) = y_{0}$$ and $$u_{2}(0) = y'(0)$$; the latter which you do not specify in your original problem statement. But looking at the second equation, it would need to be zero since otherwise the last term gets a division by zero.
Next, you would need a routine that returns $$U(x)$$ as a function of $$x$$. You say that you have no analytical form but only numerical data. So you need to write a routine that interpolates the available numerical data (or an another type of approximation you see to be fit).
Then, you could launch a series of calculations with different values of your parameter a to see when the solution tends to zero for $$x\to\infty$$. And wrap these in a secant solver (or any other solver) to find values for a.
• Thank you very much for your response. Basically the step that I had no idea of how to proceed is about the routine for the interpolation for $U(x)$. I will try to continue with this new information and I will let you know if I have any problem. Again, thank you very much for your reply! – Luis Enrique Padilla Albores Apr 13 at 10:50


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# A set intersecting the graph of any continuous function in a finite set

New version of the problem I am looking for a characterization of those completely regular and hausdorff spaces $X$ such that the follwing is true:

If $f :X\longrightarrow \Bbb{R}$ is continuous such that for any continuous function $g: X \longrightarrow \Bbb{R}$, $\mbox{Graph}(g)\cap A_f$ is finite, then $A_f$ is finite. (where $A_f := \{ (x, \frac 1{f(x)}) \; | \; f(x)\not = 0 \}$).

As @NikWeaver has pointed out below this is now true in general.

Solution to the new version: $X$ has this property if and only if it does not contain a countably infinite set of isolated points every infinite subset of which has a cluster point.
In one direction, if $X$ does contain such a configuration, then the counterexample given in my other answer generalizes straightforwardly. Conversely, suppose it contains no such configuration and suppose $f: X \to \mathbb{R}$ is continuous and $A_f$ is infinite. There are two possibilities. First, suppose that for some $\epsilon > 0$ the set $\{x \in X: |f(x)| \geq \epsilon\}$ is infinite. Then find a continuous function $\phi: \mathbb{R} \to \mathbb{R}$ such that $\phi(t) = \frac{1}{t}$ for $|t| \geq \epsilon$. The function $g = \phi \circ f$ then has $\Gamma(g) \cap A_f$ infinite, so $f$ is not a counterexample. Otherwise, suppose that the set $\{x \in X: |f(x)| \geq \epsilon\}$ is finite for every $\epsilon > 0$. By the assumption on $X$ we can find an infinite subset $S$ of $\{x: f(x) \neq 0\}$ that has no cluster point. Then define $g(x) = \frac{1}{f(x)}$ when $x \in S$ and $g(x) = 0$ when $x \not\in S$. Then $g$ is continuous and $\Gamma(g) \cap A_f$ is infinite, so again $f$ is not a counterexample.
• Incidentally, this condition is weaker than "does not contain a countably infinite set of isolated points that has a limit point". For instance, let $X$ be $\mathbb{N}$ together with one point of $\beta \mathbb{N}$. This $X$ does contain a copy of $\mathbb{N}$ that has a cluster point, but every infinite subset of $\mathbb{N}$ can be split into two infinite pieces, only one of which has a cluster point. – Nik Weaver Sep 30 '13 at 22:59
• @LisaK: the generalization is this. If $X$ contains such a configuration, then identify it with ${\bf N}$ and define $f$ as in my other answer. Let $f$ be zero off the configuration. A cluster point of a set $A$ is a point $x$ every neighborhood of which has infinite intersection with $A$. – Nik Weaver Jan 10 '15 at 14:27
No. Let $X$ be the one-point compactification of $\mathbb{N}$ and define $f(n) = \frac{1}{n}$, $f(\infty) = 0$. Then $A_f$ is infinite but $\Gamma(g) \cap A_f$ is finite for any continuous $g$ since continuous functions are bounded.


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## DOE Issues $6M Solicitation for On-Board Vehicular Hydrogen Storage R&D ##### 29 March 2006 The Department of Energy (DOE) Office of Energy Efficiency and Renewable Energy (EERE) has issued a solicitation for applied research and development projects on hydrogen storage technologies for on-board vehicular applications. EERE is seeking applications in two categories. The first includes projects supportive of and complementary to the activities of any of the existing Hydrogen Storage Centers of Excellence in Metal Hydrides, Chemical Hydrogen Storage, and Carbon-Based Materials. Projects in this category must help establish important new technical approaches or capabilities not presently available at the Centers. A funded project may become a Center partner. The second is independent R&D projects that address one of three technical topics: 1) Materials Discovery; 2) Engineering Science; or 3) Systems, Safety and Environmental Analyses. EERE is not seeking projects involving cylindrical high pressure or liquid on-board storage tanks and off-board storage, and will not review such for this solicitation. Total funding available for the solicitation is$6 million, with a per-award cap of $2 million. For transportation, the overarching technical challenge for hydrogen storage is how to store the amount of hydrogen required for a conventional driving range (greater than 300 miles), within the vehicular constraints of weight, volume, efficiency, safety, and cost. (Earlier post.) The key technical challenges, according to the DOE, for all approaches of vehicular hydrogen storage include: • System Volume and Weight. The volume and weight of current hydrogen storage systems are too high, resulting in inadequate vehicle range. • System Cost. The cost of on-board hydrogen storage systems is too high, particularly in comparison with conventional storage systems for petroleum fuels. • Efficiency. Energy efficiency is a challenge for all hydrogen storage approaches. The energy required to get hydrogen in and out is an issue for on-board reversible materials. Life-cycle energy efficiency is a challenge for chemical hydrogen storage in which the by-product is regenerated off board the vehicle. Thermal management for charging and releasing hydrogen from the storage system needs to be optimized to increase overall efficiency for all approaches. • Durability and Operability. Durability of hydrogen storage systems is inadequate. Storage media, materials of construction and balance-of-plant components are needed that allow hydrogen storage systems with a lifetime of at least 1,500 cycles and with tolerance to hydrogen fuel contaminants. An additional durability issue for material-based approaches is the delivery of sufficient quality hydrogen for the vehicle power plant. • Charging and Discharging Rates. Hydrogen refueling times—especially for material-based systems—are too long. DOE wants to see hydrogen storage systems with refueling times of less than three minutes for a 5kg of hydrogen charge, over the lifetime of the system. Thermal management that enables quicker refueling is a critical issue that must be addressed. Also, all storage system approaches must be able to supply sufficient flow rate of hydrogen to the vehicle power plant (e.g. fuel cell or internal combustion engine) to meet the required power demand. • Thermal Management. In general, the main technical challenge is heat removal upon re-filling of hydrogen for on-board reversible materials within fueling time requirements. On-board reversible materials typically require heat to release hydrogen on board the vehicle. Heat must be provided to the storage media at reasonable temperatures to meet the flow rates needed by the vehicle power plant, preferably using the waste heat of the power plant. Depending upon the chemistry, chemical hydrogen approaches often are exothermic upon release of hydrogen to the power plant, or optimally thermal neutral. By virtue of the chemistry used, chemical hydrogen approaches require significant energy to regenerate the spent material and by-products prior to re-use; this done off the vehicle. • Codes & Standards. Applicable codes and standards for hydrogen storage systems and interface technologies have yet to be established. Standardized hardware and operating procedures, and applicable codes and standards, are required. • Life-Cycle and Efficiency Analyses. Systematic analyses for the full life-cycle cost, efficiency, and environmental impact for hydrogen storage systems are required. Additional issues specific to reversible material-based hydrogen storage systems (i.e. materials that may be charged and discharged reversibly on board a vehicle) are: • Lack of Understanding of Hydrogen Physisorption and Chemisorption. An improved understanding of the fundamentals and optimization of adsorption/absorption and desorption kinetics are needed to optimize hydrogen uptake and release capacity rates. An understanding of chemical reactivity and material properties, particularly with respect to exposure under different conditions (air, moisture, etc.) is also lacking. • Reproducibility of Performance. Standard test protocols for evaluation of hydrogen storage materials are lacking. Reproducibility of performance both in synthesis of the material/media and measurement of key hydrogen storage performance metrics is an issue. Standard test protocols related to performance over time such as accelerated aging tests as well as protocols evaluating materials safety properties and reactivity over time are also lacking. Additional issues specific to chemical hydrogen storage systems (i.e. materials that may discharge hydrogen on board but need to be regenerated off board) are: • Regeneration Processes. Low-cost, energy efficient regeneration processes have not been established. Full life-cycle analyses need to be performed to understand cost, efficiency and environmental impacts. • By-Product/Spent Material Removal. he refueling process is potentially complicated by removal of the by-product and/or spent material. System designs must be developed to address this issue and the infrastructure requirements for off-board regeneration. DOE Targets for on-Board Hydrogen Storage Systems (ICE and fuel cell, Range >300 miles) Parameter Units 2007 2010 2015 System Gravimetric Capacity (specific energy) kWh/kg (kg H2/kg system) 1.5 (0.045) 2.0 (0.060) 3.0 (0.090) Volumetric Capacity (energy density) kWh/L (kg H2/L system) 1.2 (0.036) 1.5 (0.045) 2.7 (0.081) Storage System Cost$/kWh net
($/kg H2)$/gge at pump
6
(200)
4
(133)
2–3
2
(67)
2–3
Operating ambient temperature ºC -20/50 -30/50 -40/60
Min/max delivery temperature ºC -30/85 -40/85 -40/85
Cycle life variation % of mean (min)/ % confidence N/A 90/90 99/90
Cycle life (¼ tank to full) Cycles 500 1,000 1,500
Min delivery pressure from tank Atm (abs) Fuel cell
ICE
8
10
4
35
3
35
Max delivery pressure Atm (abs) 100 100 100
System fill time min 10 3 2.5
Minimum full flow rate (g/s)/kW 0.02 0.02 0.02
Start time to full flow (20ºC) s 15 5 5
Start time to full flow (-20ºC) (g/s)/kW 30 15 15
Start time to full flow (20ºC) s 15 5 5
Transient response (10%-90% and 90%-0% s 1.75 0.75 0.75
Fuel purity (H2 from storage) % H2 99.99 (dry basis)
Loss of useable H2 (g/h)kg H2stored 1 0.1 0.05

Resources:

So many rocket science involved, they will need decades or even centuries to solve all those problems. And at the end energy market is control by technology giants, everyone still need to pay they huge hydrogen fee to live.

Of course, this is one of the vision. Another kind of vision is everyone can generate and refill hydrogen at home. Convinience, clean, efficient, and economy.

Maybe when the decade comes around for the at home hydrogen, we will maybe in the cashless pleasant society.

Start time to full flow (20ºC) s 15 5 5

OK, what is wrong with methanol?

Robert -

Methanol is fairly cheap to produce (from natural, biogenic or synthesis gas), is liquid and has a high octane rating of ~120.

The downsides are:
- poor energy density by volume (better than hydrogen but worse than nethanol)
- poisonous (more so than ethanol)
- requires highly resilient fuel system components (more so than ethanol)
- flame has no colour (spills are far more dangerous)

It could make sense to blend methanol into gasoline in low proportions, especially since ethanol production capacity is still limited. On problem is that current gasoline specifications sharply limit the amount of oxygen (in any compound) it may contain.

6 million is essentially non-news.

I was watching TV last night and caught this news:

"Total sales of more than $100 million was reported at the 2006 Barrett-Jackson 35th Anniversary Auction held Jan. 17-22, its swan song at WestWorld in Scottsdale." (also here) So tell me again about 6, 15, whatever, million dollar projects. This is PRETTY good news but... 6 million??? That's chump change. Clearly the DOE doesn't care about our environment nor economy. And obviously they are in big oil's back pocket. The oil companies will probably be the ones who will lobby to receive the$6 million award money for their alternative energy research programs for good public relationship image. They will then actually develop some good solutions that actually may help solve the problems, but then patent the ideas and sit on the new technology until their patents run out in 20 years.

God bless America!

I agree with a couple of the other posters. 6 mil is nothing in comparison to other types of grants for research. The Commerce Dept. had 100 mil available for PBS stations converting from analog to digital! Unfortunately this amounts to lip service and not much of that!

Big oil should be pouring  into r&d for alternative energies. As an industry their profit was more than 50 billion in 2005 alone. That was profit after all their research into locating oil reserves and r&d for squeezing oil from shale, etc. Alex (above) was probably right. They have the funds needed to do productive research and when they identify breakthoughs they will sit on them until profit margins are sky high.

Universities with their shrinking funding sources and small alternative energy companies do not have the resources to make large scale inroads into this area.

6 mil is hardly even a start. But now is the time to really push this agenda not later when we as a nation (USA) or the world is at a critical stage of energy need and energy produced.

In line with the on going research and development on On-board Vehicular Hydrogen Storage (which I may term as: Part of the steps to 21st Century Glogal Solution); Iam making research on the use of metal hydride and (or) the use of cutting-edge polymer (a carbon-base material).
Sincerely speaking, there are inadequaces in the area of research facilities that could be at one's best disposal for effectiness in this kind of research.
Please how could you help in getting me partner to good Research and Development Centre(s) so as to jointly share ideas of mine and thereafter proceed in this laudable research goal together. Kindly mail your reply via my box. Thank you.

The comments to this entry are closed.


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# How to avoid violating energy conservation when making shaders and node groups?

Recently a user on here offered a tip to be careful when making shaders so I don't violate energy conservation. Blender stores colors in the value range of 0-1. If you go above 1, then more light is being reflected than is being received and you get physically inaccurate results.

In order for materials to work well with global illumination, they should be, speaking in terms of physics, energy conserving. That means they cannot reflect more light than comes in. This property is not strictly enforced, but if colors are in the range 0.0 to 1.0, and BSDFs are only mixed together with the Mix Shader node, this will automatically be true.

So my question is, what scenarios would break energy conservation (other than what is mentioned in the documentation)? When do I need to be vigilant about this?

I've been making utility node groups for myself - an example is my Falloff Map node shown below:

And this "texture adjustment" node group:

I'm a little concerned that my node groups could be outputting values higher than 1. For instance in the falloff map node group, I have an exponential "power" math node that then feeds into an RGB Curves node. Wouldn't the power node have the potential to output values above 1?

Another example with my "Texture Adjust" node group: I have three MixRGB nodes set to screen chained together for an alternate way to lighten a texture than simply adjusting brightness.

Is it possible that I am violating energy conservation with either of these node groups?

I am also aware of the ability to clamp values, but I usually leave it unchecked. Should I be clamping values as I go about building node groups?

I've attached both node groups so you can download and test them yourself.

Falloff Map nodegroup:

Thanks so much for any advice or help.

• Hello :). This whole thread on devtalk is interesting: Principled BSDF violating energy conservation? – Jachym Michal Feb 23 at 21:24
• And here's the 2012 paper from Brent Burley introducing the Principled BRDF shader. The Principled BRDF is leaning toward being more artist-friendly than strictly energy-conserving. – Jachym Michal Feb 23 at 21:27
• Thank you, I'll read through those links! Appreciate it. :) – Jay Feb 23 at 21:42
• @JachymMichal With respect to my node setups above, do you think I am violating energy conservation? I'm trying to understand what circumstances or nodes I need to be careful with. Or should I just clamp at the very end always? – Jay Feb 23 at 21:47

It depends on how you're going to use the value.

Generally, anything that goes into a shader should be within the 0 - 1 range.

But not necessarily Normal values or perhaps Subsurface (theoretically in either case your model would have to be huge for this to make sense) because these values deal with scale or distance.

The same could be said for the Emission values, which turn the surface of your mesh into a light source.

When creating a mask you might generate values far outside the 0 - 1 range, but this may not be a problem. For example, if you want to generate a displacement map for a mountain range you need to generate bumps several hundred meters high. If, however, this has anything to do with Base Color or any physical property of the material, you will want to do something to normalize the value.

So what can you do?

You can check the Clamp button, this takes any value above 1 and reports 1 instead.

Or, if you know the new minimum and maximum values you're dealing with (you did the math), you can use a Map Range node to compress that larger range back into the 0 - 1 range.

Regarding when to clamp:

It depends, again, on what you want. Consider the following example which is a gradient with a value from 0 to 10.

And then say we clamp the multiply.

We have almost all of this plane with a value of 1. Now if we wanted to move this gradient over some we might subtract something from this, let's say .6

Well we've moved it but now the majority of the plane has a value of .4, and to boot it appears the portion of the plane that actually had a gradient has shrunk significantly.

That's not actually true because what we've done is create a gradient of negative values on the left all of which are represented by black. A whole other can of worms!

So probably what we want to do here is not clamp the multiply, so the subtract operation has some higher values to deal with, and then do clamp the subtract so that we don't generate negative values.

Of course, all according to what you want. If you want the short gradient ending at .4 you should clamp both!

So what are some specific things to watch out for?

You are now armed with this knowledge, you can never unlearn it. With regard to color you should now be afraid of basically any unclamped Mix operation aside from good old fashioned Mix (and that's only if you know the colors going in are safe in the first place), physical properties outside 0 - 1, and then specifically the Add Shader. 👍

Any times where I shouldn't clamp?

When you're operating on the coordinate space itself, you are necessarily operating on values above 1 and in the negative range. These operations shouldn't be clamped unless there is some particular behavior you're going for.

It may seem like a lot to maintain, but now you know.

• I would point out that emission can be any value you want, not limited to 0-1 – susu Feb 24 at 2:29
• This is an incredible post. Thank you so very much for the breakdown. Wish I could give you more rep! – Jay Feb 24 at 22:46
• What I've found is that doing these is a good way for me to organize and reinforce my own thoughts about things. Plus, you know... it's one way to leave a mark on the world. – Allen Simpson Feb 24 at 22:53

It is impossible to say whether the groups you show conserve energy, because whether or not they conserve energy depends on what values you send them, and they don't output shaders anyways.

If you want to conserve energy, and you don't want to clamp unless you have to, there are only various things to know:

1. If you send a shader a color outside of the 0,1 range of color, you probably don't conserve energy.
2. If you use emission, you don't conserve energy. (Emission creates energy from nothing.)
3. If you use an add shader node, you probably don't conserve energy.
4. If you use specular values outside of the 0,1 range in a principled shader, you probably don't conserve energy.
5. If you use sheen or clearcoat in a principled shader, you probably don't conserve energy. (Not 100% sure about sheen, but 100% sure about clearcoat.)
6. If you send a mix shader a fac outside the 0,1 range, you probably don't conserve energy.

That's a lot of "probably"s. Because there are corner cases. Is it safe to add shader a 1,0,0 diffuse to a 0,1,1 diffuse? Yes. Because the output cannot exceed 1,1,1. Is it safe to add 0.1 clearcoat to a 0.9 diffuse principled? Yes, because the output cannot exceed 1,1,1. Is it safe to mix shader a 0.5 diffuse shader at fac 2.0 from a 0.0 diffuse shader? Yes, because you cannot exceed 1,1,1.

I'm sure I could have missed some examples.

• I want to start pushing back against emission not conserving energy. I think instead we should say if you are using emission you are turning the mesh into a light. It's like trying to use the concept of energy conservation when it comes to point or area lights. It's one of the things that can give your scene energy. You would only have to potentially justify it from an artistic perspective with a power source. – Allen Simpson Feb 24 at 0:52
• But this is a great post, I don't want you to take this the wrong way. :) – Allen Simpson Feb 24 at 0:55
• @Allen Simpson I don't disagree with you. Considering emission (or, for that matter, lighting) as failing to conserve energy is a technical consideration-- the reality is, pure, whole-scene energy "conservation" isn't actually a desirable characteristic! Emission doesn't conserve energy, but it's not like it would make any sense to have emission that did. Note that this does actually have some consequences and meaning, for rasterizers that aim for energy conservation (where emission is not light), for using emission to cheat things like volume scatter, and surfaces that reflect and emit. – Nathan Feb 24 at 5:12

You use clamp checkbox

It limits outputs to only numbers between 0 and one

There is also a clamp node that does the same thing, but you arent limited to only 0-1 with the node

• Like I said in my post, I do know what clamping does, but I'm trying to figure out when I absolutely need to clamp, and when I don't. Also, with more complex node setups I think clamping could result in inaccurate values if you're not paying attention. Should I only clamp at the very end? – Jay Feb 23 at 21:44


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# Running of $α s$ in the MSSM with three-loop accuracy - High Energy Physics - Phenomenology

Running of $α s$ in the MSSM with three-loop accuracy - High Energy Physics - Phenomenology - Download this document for free, or read online. Document in PDF available to download.

Abstract: The evolution of the strong coupling constant $\alpha s$ from $M Z$ to theGUT scale is presented, involving three-loop running and two-loop decoupling.Accordingly, the two-loop transition from the $\bar{ m MS}$ to the $\bar{ mDR}$ scheme is properly taken into account. We find that the three-loop effectsare comparable to the experimental uncertainty for $\alpha s$.

Author: Luminita Mihaila

Source: https://arxiv.org/


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Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 Bib Ind

### 2 Installation

datastructures does not work without compiling its kernel module, and is not loaded by GAP by default. To load the package run LoadPackage("datastructures"); at the GAP prompt.

#### 2.1 Building the Kernel Module

To build the kernel module, you will need

• a C compiler, e.g. GCC or Clang

• GNU Make

To install a released version of this package, extract the package's archive file into GAP's pkg folder.

To install the current development version of this package, obtain the most recent code from

git clone https://github.com/gap-packages/datastructures

To build the kernel module then run the following commands in the package's directory.

./configure
make

#### 2.2 Building the Documentation

To build the package documentation, run the following command in the package's directory

gap makedoc.g

Goto Chapter: Top 1 2 3 4 5 6 7 8 9 10 11 Bib Ind

generated by GAPDoc2HTML


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# Beryllium 7 dating tony aussie dating

For information on each type of decay, read the page Decay pathways.In terms of entropy, radioactive decay can be defined as the tendency for matter and energy to gain inert uniformity or stability.Also, radioactive decay is an exponential decay function which means the larger the quantity of atoms, the more rapidly the element will decay.Mathematically speaking, the relationship between quantity and time for radioactive decay can be expressed in following way: $\dfrac = - \lambda N \tag$ or more specifically $\dfrac = - \lambda N \tag$ or via rearranging the separable differential equation $\dfrac = - \lambda dt \tag$ by Integrating the equation $\ln N(t) = - \lambda t C \tag$ with There are two ways to characterize the decay constant: mean-life and half-life. As indicated by the name, mean-life is the average of an element's lifetime and can be shown in terms of following expression $N_t=N_o e^ \tag$ $1 = \int^_ 0 c \cdot N_0 e^ dt = c \cdot \dfrac \tag$ Rearranging the equation: $c= \dfrac$ Half-life is the time period that is characterized by the time it takes for half of the substance to decay (both radioactive and non-radioactive elements).

There are five types of radioactive decay: alpha emission, beta emission, positron emission, electron capture, and gamma emission.

The atomic nucleus which is in the center of the atom is buffered by surrounding electrons and external conditions.

Because of this, the study of decay is independent of the element's environment.

Radioactive decay is not always a one step phenomenon.

Often times the parent nuclei changes into a radioactive daughter nuclei which also decays.

### Search for beryllium 7 dating:

However, there exists a lighter isotope of Beryllium which contains 4 protons and only 3 neutrons, which gives a total mass of 7 amu.

## One thought on “beryllium 7 dating”

1. It was a long wait — we are both in our mid-40s — but well worth it. I had never dated anyone with children, and I never wanted my own. She’d called when we were driving to the Mark Taper Forum in downtown Los Angeles, and we put her on speakerphone while I remained silent because she didn't yet know her dad was dating.


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# How Does a Vacuum Tube Work

### Two key concepts for understanding how a vacuum tube works are thermionic emission and charge attraction and repulsion.

Vacuum Tube / Thermionic Valves Includes:
Basics     How does a tube work     Vacuum tube electrodes     Diode valve / tube     Triode     Tetrode     Beam Tetrode     Pentode     Equivalents     Pin connections     Numbering systems     Valve sockets / bases

The theory behind the operation of a vacuum tube is based on a concept known as thermionic emission.

In addition to this concepts including the attraction and repulsion of opposite and like charges play a large part in the operation of vacuum tubes / thermionic valves.

Understanding these concepts provides the basis behind understanding how a vacuum tube works.

## Thermionic emission

The first concept needed to understand how a vacuum tube works is that of thermionic emission.

Electrical conductivity of metals results from the fact that there are free electrons moving around the material and not attached to any definite molecule. Although there are equivalent numbers of holes so that the overall charge remains balanced, these electrons roam freely around the material.

If these electrons are to leave the surface of the material, work needs to be done to overcome the attraction within the material.

The energy required to overcome the forces holding the electrons within the material can be supplied in a number of ways. One of these is to heat the material and in this way the electrons receive additional kinetic energy. At a sufficiently high temperature, some electrons will have sufficient kinetic energy to escape from the surface of the material. This is thermionic emission of electrons, and it is this phenomenon that is at the core of how a vacuum tube works.

The process of thermionic emission from a material has many similarities with that of evaporation from the surface of a liquid. In the case of molecules in a liquid, the ones that escape and evaporate have sufficient energy to escape the retraining forces of the liquid, and the number escaping increases with increasing temperature. It can be considered as essentially the same process in which the energy the electron must give up corresponds to the latent heat of vaporisation in a liquid.

## Electron emission

In looking at how a vacuum tube works it is also necessary to consider the effectiveness of the way in which electrons escape from the surface.

The number of electrons emitted from the heated material per unit area is related to the absolute temperature as well as a constant 'b' that is a constant indicating the work an electron has to do to escape the surface.

As a result it is possible to derive an equation for the current leaving the surface:

$I=A{T}^{2}{\epsilon }^{\left(}-b/T\right)$

Where:
I = current measured in Amperes
A = a constant for the type of emitting material
T= temperature in degrees Absolute
b = work required for electron to leave surface

## Electron emitters - cathode materials

It is necessary to reach temperatures in excess of 500°C, dependent upon the material, for the number of electrons leaving the surface of the material to become appreciable. When working with temperatures of this order, it limits the materials that can be used on the cathodes of vacuum tubes.

There are a few favoured emitters that are used within vacuum tubes:

• Tungsten:   Tungsten provides one of the most robust forms of filament for a vacuum tube, particularly when very high anode voltages are used. However its drawback is that its emission efficiency expressed in terms of amperes emission per watt of heating is not as high as other emitters like thoriated tungsten and oxide coated emitters.
• Thoriated tungsen:   Thoriated tungsten is widely used in vacuum tubes and consists of tungsten containing 1 to 2% of thorium oxide. Vacuum tubes / thermionic valves using cathodes with this coating give electron emission at temperatures of between 1500° and 1600°K. Vacuum tubes using thoriated tungsten must have a very high degree of vacuum otherwise the positive ions produced by ionisation of gases in the envelope will seriously affect the emission.
• Oxide coated emitters:   Vacuum tubes using this form of cathode coating have layer of a mixture of barium and strontium oxides coating the surface of the cathode. When properly activated they emit electrons profusely at a temperature of around 1100° to 1200°K. Oxide coated emitters are widely used because they give more emission per watt of heating than any other type. One drawback is that the emitting surface is easily poisoned by impurities. Vacuum tubes using oxide coatings are used for most small vacuum tubes / thermionic valves using voltages up to a few thousand volts.

Although, normally vacuum tubes are indirectly heated these days, this form of heating is less efficient than the directly heated option. As a result, some specialist tubes or valves that use tungsten or thoriated tungsten filaments sometimes use direct heating techniques.

## Space charge

One important aspect of vacuum tube theory is the space charge.

The electrons flowing between the cathode and the anode form a cloud of electrons and this is known as the "space charge". The space charge tends to repel electrons leaving the cathode, forcing them back. However if the potential applied to the anode is sufficiently high then the space charge effect will be overcome, so that electrons will flow toward the anode. In this way electrons are able to move across the vacuum within the glass envelope of the vacuum tube / valve, the circuit is completed and current flows.

As the potential is increased on the anode, so the current increases. Eventually a point is reached where the space change is completely neutralised and the maximum emission from the cathode is attained. The only way in which the electron emission fromt the cathode can be increased is by increasing the cathode temperature. This increases the energy of the electrons and as a result it allows further electrons to leave the cathode.

Although all areas of a vacuum tube have a space charge, it is of particular importance in the cathode region as it determines elements including the maximum emission.

As other electrodes are added into the evacuated envelope, the space charge concept can be applied to the whole operational area.

The space charge concept plays a crucial role in determining the flow of current in any thermionic device.

## Child's Law

Child's Law, often also called the Child-Langmuir Law was first proposed in 1911 and it forms a key elements within thermionic valve or vacuum tube theory and how a vacuum tube works.

Child's Law states that the space-charge limited current in a plane-parallel vacuum diode varies directly as the three-halves power of the anode voltage and inversely as the square of the distance d separating the cathode and the anode.

$J=\mathrm{Ia}/S$

Where:
J = current density in Amperes per metre squared,
Ia = anode current,
S = anode surface area in square metres

Child derived this equation applicable to vacuum tube theory in 1911 for the case of atomic ions. These have much smaller ratios of their charge to their mass. Irving Langmuir extended the basic law when he published the application to electron currents in 1913. This extended it to the case of cylindrical cathodes and anodes. It is for this reason that the law is sometimes referred to as the Child-Langmuir Law.


text:
# Graph each polynomial function.f(x)=2x^{3) - 7x^{2} + 2x + 3given that 3 is a zero.

Graph each polynomial function. given that 3 is a zero.

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Velsenw

Step 1 To find the y-intercept evalute f(0)
Substitute 0 for x $=3$ Simplify So, the y-intercept is (0, 3)

Step 2 given that 3 is a zero Use synthetic division The quotient is

So, are zeros of f(x) The graph of f(x) crosses the x-axis at Plot the points See the graph below.