text:
Journal cover Journal topic
Annales Geophysicae An interactive open-access journal of the European Geosciences Union
Journal topic
Ann. Geophys., 36, 1073-1080, 2018
https://doi.org/10.5194/angeo-36-1073-2018
Ann. Geophys., 36, 1073-1080, 2018
https://doi.org/10.5194/angeo-36-1073-2018

Regular paper 08 Aug 2018

Regular paper | 08 Aug 2018

# Statistical survey of day-side magnetospheric current flow using Cluster observations: bow shock

Statistical survey of bow shock currents
Evelyn Liebert, Christian Nabert, and Karl-Heinz Glassmeier Evelyn Liebert et al.
• Institut für Geophysik und extraterrestrische Physik, Technische Universtität Braunschweig, Braunschweig, Germany
Abstract

We present the first comprehensive statistical survey of the day-side terrestrial bow shock current system based on a large number of Cluster spacecraft bow shock crossings. Calculating the 3-D current densities using fluxgate magnetometer data and the curlometer technique enables the investigation of current locations, directions, and magnitudes in dependence on arbitrary IMF orientation. In case of quasi-perpendicular shock geometries we find that the current properties are in good accordance with theory and existing simulation results. However, currents at quasi-parallel shock geometries next to the foreshock region underlie distinct variations regarding their directions.

1 Introduction

The terrestrial bow shock slows down the solar wind velocity to subsonic Mach numbers. This is accompanied by a gain in density, temperature, and magnetic field strength. According to Ampère's law the bow shock carries electric currents which account for the jump in the magnetic field components tangential to the bow shock's surface. In contrast to the magnetopause, where the current directions are mainly determined by the geometry of the Earth's magnetic field, the direction of bow shock currents is solely determined by the orientation of the interplanetary magnetic field (IMF).

Depending on the local geometries of the shock surface and IMF orientation one distinguishes between quasi-perpendicular and quasi-parallel shocks where the IMF encounters the bow shock normal with angles above and below 45, respectively. When encountering the compressed magnetic field at the shock some solar wind particles are reflected at the shock and re-accelerated in the solar wind's electric field while gyrating along the IMF direction before they enter the shock another time. At the quasi-parallel shock the reflected particles form the foreshock region where upstream waves are generated and alter the magnetic field configuration.

To date, detailed bow shock current analysis based on in situ measurements has barely been done. presented a first statistical survey of bow shock currents using Cluster data from 25 crossing events when the IMF was dominated by its Bz component. They selected quasi-perpendicular shocks near the bow shock nose and calculated the current density from the magnetic gradient and the shock thickness. Recently, investigated the currents of 154 quasi-perpendicular bow shock crossings near the shock nose using data from the Magnetospheric Multiscale (MMS) mission. In this paper we extend the statistical survey of bow shock currents at larger distances to the bow shock nose to overall 369 events covering both quasi-perpendicular and quasi-parallel situations during arbitrary IMF configurations for the first time. Making use of the simultaneously collected magnetic field data supplied by the Cluster multi-spacecraft mission and applying the curlometer technique allow a direct 3-D investigation of the local current density vector.

2 Data selection and preparation

For our investigation we use Cluster magnetic field data from the fluxgate magnetometer (FGM) at spin resolution (0.25 Hz). Additionally, data from the Cluster Ion Spectrometry (CIS) instrument are used to support the identification of bow shock crossings. The data are retrieved from the Cluster Active Archive . Cluster consists of four individual spacecraft orbiting along a polar orbit with relative separations of a few kilometers up to over 10 000 km. The thickness of the Earth's bow shock is about 100 to 1000 km. In order to match these spatial dimensions we use data obtained during periods when the average inter-spacecraft distance was small at the position of the bow shock. This criterion is fulfilled in the time range from February to May 2002 and from December 2003 to May 2004 when the average inter-spacecraft distance is about 300 km or less. At these times FGM and CIS data are available at the bow shock for 162 inbound and outbound orbit segments.

The curlometer technique estimates the local current density across the Cluster tetrahedron volume by approximating Ampère's law: $\mathrm{\nabla }×\mathbit{B}={\mathit{\mu }}_{\mathrm{0}}\mathbit{J}$. A thorough introduction to the curlometer technique and an error analysis focussing on our application can be found in our previous publication , where we conducted a similar study on magnetopause currents. The reliability of curlometer results depends on the Cluster tetrahedron geometry which is constantly changing along its trajectory. One possibility to characterize the shape of the tetrahedron is given by the quality factor QG, which is defined by

$\begin{array}{}\text{(1)}& {Q}_{G}=\frac{\text{true\hspace{0.17em}\hspace{0.17em}volume}}{\text{ideal\hspace{0.17em}\hspace{0.17em} volume}}+\frac{\text{true\hspace{0.17em}\hspace{0.17em} surface}}{\text{ideal\hspace{0.17em}\hspace{0.17em} surface}}+\mathrm{1}\end{array}$

(Glassmeier et al.2001), where the ideal volume and surface represent the volume and surface of a perfect regular tetrahedron with a side length equal to the average side length of the true tetrahedron. The quality factor equals 1 when the true tetrahedron is deformed into a linear geometry and 3 in case the true tetrahedron equals the ideal one. In this study we limit our investigation to bow shock crossings, where the quality factor takes values of 2.5 or more, leaving us 111 orbit segments. The usage of this quality factor for our investigation is discussed in detail by . QG≥2.5 allows us to expect accuracies of at least 2 to 10 in direction and 3 % to 15 % for the relative error in magnitude.

Figure 1This sketch of the data selection process shows the shrinking number of suitable data sets. From over 2500 Cluster orbits only a few orbit segments match the spatial requirements and the quality criterion for our bow shock current investigation.

pointed out that high-frequency fluctuations are likely to cause uncertainties within the curlometer results and suggest an appropriate averaging in time before applying the curlometer to magnetic field data. Comparing the effect of averaging windows of different sizes on the events investigated in our study proves the current directions to be quite insensitive to a window size between 20 and 40 s. Significant alteration of the directions sets in for windows below 10 s and above about 60 s as the influence of spatial scales smaller and larger than the bow shock scales rises. Current magnitudes are more sensitive to the size of the averaging window. The damping of high-frequency fluctuation and the associated spatial averaging directly lead to smaller current peak magnitudes. This effect is less intense when an average magnitude per event is calculated instead. For the statistical study presented here, we chose a 30 s averaging window, which proved to be sufficient to damp highly fluctuating current signatures with spatial dimensions far below the tetrahedron size without altering current structures having dimensions of about 50 km and more along the spacecraft trajectory to a significant extent.

Figure 2Example of a bow shock crossing during an outbound orbit on 11 February 2004. The upper and middle panels show Cluster 1 magnetic field data and the curlometer results in GSE coordinates after application of a 30 s averaging window. The lower panel shows Cluster 1 CIS hot ion density n, velocity v, and temperature T. At 16:08 UT the transition from magnetosheath plasma (left-hand side) into undisturbed solar wind plasma (right-hand side) is visible within the particle data as well as within the magnetic field data. During the bow shock crossing a clear current density peak is seen.

After the application of the curlometer tool, we look for bow shock current events by visual inspection of the curlometer results. Currents are identified as bow shock currents when the following criteria are fulfilled: (1) a clear current peak is visible, and (2) the current event coincides with particle data signatures that are consistent with a bow shock crossing (see the example event in Fig. 2). At each transition event the edges of the corresponding current feature are identified. We calculate the average current directions and magnitude for every event. There are cases where the current or the particle data or both show very fluctuating behavior, making it difficult to identify current structures at a bow shock crossing. In cases where the identification becomes presumably unreliable, the events are omitted and not included in our study.

Caused by relative movement of the bow shock with respect to Cluster, multiple crossings in a row are often recorded along an orbit segment. This enlarges our database for the statistical survey to 369 current events. Figure 3 gives an overview of the locations of all events in GSE coordinates.

Figure 3Location of the investigated bow shock crossings projected onto the reference bow shock presented in a GSE coordinate system. The grey ellipsoids represent the bow shock position at $x=\mathrm{0.2},\mathrm{0.5}$, and 0.8ΔBS with ΔBS depicting the bow shock stand-off distance.

3 Reference bow shock

The position and size of the bow shock vary depending on the solar wind conditions to a large extent. For representation of the bow shock crossing locations we introduce a parabolic model reference bow shock as a common frame of reference. Following we use the parametrization

$\begin{array}{}\text{(2)}& x={\mathrm{\Delta }}_{\mathrm{BS}}-\sum _{t=y,z}{c}_{\mathrm{BS},t}\phantom{\rule{0.125em}{0ex}}{t}^{\mathrm{2}}.\end{array}$

ΔBS depicts the sub-solar bow shock stand-off distance with respect to the center of the Earth (see Fig. 4). The geometric parameters cBS,t represent the bow shock curvature in the t=y and t=z directions. deduce values of

$\begin{array}{}\text{(3)}& {c}_{\mathrm{BS},y}=\mathrm{0.4}\frac{\mathrm{1}}{{\mathrm{\Delta }}_{\mathrm{BS}}}\phantom{\rule{1em}{0ex}},\phantom{\rule{1em}{0ex}}{c}_{\mathrm{BS},z}=\mathrm{0.5}\frac{\mathrm{1}}{{\mathrm{\Delta }}_{\mathrm{BS}}},\end{array}$

from an analytical zeroth-order approach to solve the MHD equations in the magnetosheath.

Figure 4Sketch of the model bow shock used as a reference bow shock for data presentation. The dotted lines show example projections of two different currents from their true location to the location at the reference bow shock.

For each identified current the mean value of the tetrahedron barycenter’s position vector is calculated. By radial projection along the Earth-spacecraft line the intersection of this vector with the reference bow shock is calculated (cf. Fig. 4). Figure 3 gives an overview of the location of all events after projection onto the reference bow shock within GSE coordinates.

Current directions at the bow shock are directly controlled by the IMF orientation via Amperè's law. As we do not confine our study to mainly north–south orientated IMF, the presentation of the resulting current directions in a GSE system like in Fig. 3 would lead to a quite chaotic looking distribution and would make it almost impossible to extract useful information from it as the required information about the IMF orientation would not be included in such a picture. In order to account for the varying IMF orientations we conduct a second step of transformation by rotating the coordinate system around the GSE x axis in such a way that the IMFyz,GSE component is orientated in the positive z direction within the new IMF-aligned coordinate system. The IMF is calculated by averaging the magnetic field data obtained during 5 min ahead of each bow shock current event.

4 Results

## 4.1 Directions of bow shock currents

Depending on the angle Φ between local shock normal and IMF, current events are categorized into quasi-perpendicular ($\mathrm{\Phi }>\mathrm{45}{}^{\circ }$) and quasi-parallel shocks ($\mathrm{\Phi }<\mathrm{45}{}^{\circ }$). The majority of 274 of the investigated currents represent quasi-perpendicular geometries. It is likely that this imbalance is caused by our event selection procedure. Within the foreshock region of a quasi-parallel shock oscillations are triggered and develop while the plasma is convected towards the shock surface. This causes fluctuations within the particle and the magnetic field data, leading to less clear plasma transition and current signatures. As described in Sect. 2, events are omitted, when a reliable bow shock current identification is not possible.

Figure 5Frequency distribution of the angle difference between the current directions and the local bow shock normal (a) and the IMF tangential component (b). The angles are binned with 10 intervals.

Figure 5 shows the orientation of the current flow with respect to the model bow shock normal and the IMF. Most of the observed bow shock currents during quasi-perpendicular geometries lie nearly perpendicular to both the shock normal, which means that the currents flow parallel to the bow shock surface, and the IMF tangential component, as expected from theory. The deviation from perpendicularity with respect to IMFt is much larger in the case of quasi-parallel shock situations (95 events), which reflects more turbulent and fluctuating conditions of the plasma flow adjacent to the foreshock region in contrast to the quasi-perpendicular shock. The broad distribution of the angle between current direction and reference bow shock normal indicates that the current flow direction at the quasi-parallel bow shock deviates extremely from the shape of a simplified bow shock surface.

Figure 6Bow shock current directions in the xy plane (a) and yz plane (b). Jx direction is represented by the colour code. The z axis of the coordinate system is aligned to the IMFyz component. The grey paraboloid and ellipsoids represent the bow shock position at z=0 and $x=\mathrm{0.2},\mathrm{0.5}$, and 0.8ΔBS, respectively. Polar histograms (c) show the occurrence rates of current angles within the yz plane with respect to the positive y axis.

The global current direction distribution of the quasi-perpendicular and quasi-parallel bow shock currents is displayed in Fig. 6 in the IMF-aligned coordinate systems. The current directions are represented by arrows of normalized length and the color code indicates the direction of currents with respect to the x axis with red arrows pointing towards Earth and green arrows pointing towards the Sun. From theory, within this coordinate system the currents are expected to possess positive y components, independently of their locations. As visible in the yz projections the flow directions are prescribed by the IMF orientation very clearly as they are collectively pointing along the positive y axis in the IMF-aligned coordinate system. The histograms of the Jyz direction in the bottom panel of Fig. 6 illustrate this in a quantitative manner. It shows a very clear peak around 0 in the quasi-perpendicular case. The peak is also visible in the quasi-parallel case, but it is much broader and currents with negative y components are observed frequently.

Having a closer look at the color code and the xy projection reveals that the currents are following the draped shape of the bow shock pointing towards the Sun and Earth at the flanks. Again, more deviations are visible during quasi-parallel bow shock crossings (Fig. 6b). investigated Jx in a similar way. As their events are located near the bow shock nose because of the MMS orbit, they introduced the approximation JxJn, where Jn depicts the portion of the current that flows normal to the shock surface. Their events lie within a range of about 7 Earth radii distance from the Earth–Sun line which approximately corresponds to a distance of about 0.5 ΔBS in our coordinate system. Near the bow shock nose they find that Jx points towards the Sun at y<0 (GSM) and towards Earth at y>0 (GSM) for northward IMF. In the case of southward IMF the current directions are reversed. The distribution of the colors in Fig. 6 shows that the results of the orientation of the Jx component from the study conducted by and from our study are qualitatively identical and are also valid for larger distances from the bow shock nose.

In order to investigate the current normal component in our study, we calculate Jn via the local normal direction of the model bow shock surface for all quasi-perpendicular events. We analyze the orientation of the normal component in dependence on the y coordinate within the IMF-aligned coordinate system. Table 1 gives the occurrence rate of outward and inward orientation of Jn in a region close to the bow shock nose, approximately corresponding to the location of events within the study by as well as a region further away from the nose at the flanks of the bow shock. Based on our results, we can not identify a general dependence of the Jn orientation from the location of the events. Overall, the currents are slightly more often pointing outwards.

Table 1Occurrence rate of outward- and inward-pointing directions of the current normal component at the quasi-perpendicular bow shock depending on the y coordinate within the IMF-aligned coordinate system.

## 4.2 Current magnitudes

Figure 7 shows the occurrence distribution of investigated current magnitudes. The majority (about 80 %) do not exceed 30 nA m−2 and the average current magnitude of all events is 19.4 nA m−2. As the current magnitudes calculated by the curlometer are influenced by the averaging in time and space (averaging window, spacecraft separation, average current density along event trajectory), it is more likely that current magnitudes tend to be underestimated than overestimated. A direct comparison of some events which were analyzed in the study by as well as in our study show that the current density magnitudes calculated with the curlometer technique are factors between 2.7 and 4.5 smaller than those calculated by determination of the layer thickness and the jump in the magnetic field.

Figure 7Occurrence rate of bow shock current magnitudes with a 10 nA m−2 binning interval.

MHD simulations (Lopez et al.2011) predict a relatively broad region around the bow shock nose where the current magnitudes are constantly high, while the magnitudes are decreasing at the high-latitude bow shock. Investigating the spatial dependence of the event's current magnitudes results in a roughly homogeneous distribution in case of the quasi-parallel shock events which are all located at low latitudes (cf. Fig. 6). In case of quasi-perpendicular events current magnitudes near the bow shock nose are slightly larger than those at the flanks. The average magnitude of all quasi-perpendicular currents below 60 latitude is 22.3 nA m−2, while the average value of currents at higher latitudes is 16.1 nA m−2.

Figure 8Current magnitude in dependence of the IMF tangential component. The IMFt component is calculated via the shock normal direction and is binned into intervals of 1 nT. The current magnitude $<{J}_{\mathrm{BS}}>$ is calculated by averaging over all quasi-perpendicular and all quasi-parallel current events that are ascribed to the corresponding interval.

The IMF magnitude is another controller of the bow shock current magnitude. Based on Ampère's law and the Rankine–Hugoniot conditions one can expect a linear correlation between the current magnitude and the magnetic field strength of the IMF tangential component with respect to the shock surface

$\begin{array}{}\text{(4)}& J\propto \left[{\mathbit{B}}_{t}\right]\propto \mathbit{I}\mathbit{M}{\mathbit{F}}_{t},\end{array}$

where the brackets denote the jump of the magnetic field across the discontinuity.

found this linear relation between bow shock current and IMFz component at quasi-perpendicular bow shock events during northward- and southward-orientated IMF. Our survey shows that the linear correlation also applies to arbitrary IMF orientation and shock geometry. Figure 8 displays the average current magnitudes that are calculated for 1 nT intervals of the IMF component tangential to the bow shock surface. For quasi-perpendicular and quasi-parallel cases magnetic field values up to 15 and 8 nT are observed, respectively. Within the range from 0 to 8 nT there are no distinct qualitative differences between quasi-perpendicular and quasi-parallel situations visible.

In the limit of a high Mach number one can derive

$\begin{array}{}\text{(5)}& \mathbit{J}=\mathrm{3}\phantom{\rule{0.125em}{0ex}}\mathbit{I}\mathbit{M}{\mathbit{F}}_{t}/\left({\mathit{\mu }}_{\mathrm{0}}L\right),\end{array}$

where L is the bow shock thickness. The correlation coefficient of the linear fit within Fig. 8 is 0.84. The slope of the fit provides an estimate of the average bow shock thickness of about 1600 km. As mentioned above, it is likely that the magnitudes tend to be underestimated by the curlometer technique. The value of 1600 km therefore represents an upper estimate of the shock thickness. performed an extensive study of the bow shock thickness which gives a typical scale of a few hundreds of kilometers.

5 Conclusions

The usage of the curlometer technique allows us a direct investigation of 369 current events recorded within the magnetic field data obtained by Cluster during 111 bow shock crossings in 2002, 2003, and 2004. In 274 cases the bow shock represented a quasi-perpendicular shock ($\mathrm{\Phi }>\mathrm{45}{}^{\circ }$). In 95 events a quasi-parallel ($\mathrm{\Phi }<\mathrm{45}{}^{\circ }$) shock was observed. It lies adjacent to the foreshock region where solar wind particles are reflected back into the solar wind along the IMF direction and cause upstream and downstream perturbations of the magnetic field configuration.

At quasi-perpendicular shocks the bow shock currents are very clearly described by the IMF direction fulfilling the equation $\mathrm{\nabla }×\mathbit{B}={\mathit{\mu }}_{\mathrm{0}}\mathbit{J}$. The angle distribution between the bow shock current and IMFt shows a sharp peak at 90. When displayed in an IMFyz-aligned coordinate system the current directions' arrows arrange themselves parallel to each other in the yz plane. As the currents flow parallel to the shock surface, the draped shape of the bow shock becomes visible within the current direction in the xy plane. Current magnitudes are larger near the bow shock nose than at the flanks. Figure 9 is a schematic summary of our results for the currents observed at the day-side quasi-perpendicular bow shock.

Figure 93-D scheme of day-side bow shock current magnitude and orientation with respect to the IMF yz component (GSM) in case of a quasi-perpendicular bow shock. Greater and darker arrows represent higher values of current magnitude.

The angle distribution between currents and the normal of the reference bow shock peaks at 90 as well, but it is broadened to some extent as the reference bow shock naturally deviates from the true bow shock geometry. The magnitudes of currents at the quasi-perpendicular shock generally increase with an increasing tangential component of the IMF. Typical values of the averaged current magnitudes obtained by the curlometer technique are in the range of 5 to 40 nA m−2 with an average of about 20 nA m−2. Those results are on the same order of magnitude but smaller than the ones determined in former studies by calculating the current density via the jump in the magnetic field and a derived current sheet thickness.

The quasi-parallel shocks that we have found are all located at relatively low latitudes. In this region, we were not able to observe a spatial dependence of the current magnitude. Additionally, the IMF tangential components possess lower values only up to 8 nT. In this range we find that the dependence on the IMF magnitude seems to be qualitatively equal to that observed for quasi-perpendicular situations. In particular the current magnitudes of the quasi-perpendicular and quasi-parallel bow shocks are of a similar size for a given IMF, which is an interesting finding as the ideally (in reality never realized) parallel shock would be accompanied by no jump in the magnetic field and therefore no current at all (Narita2006).

The direction of currents of the quasi-parallel bow shock are less described by the IMF orientation compared to the quasi-perpendicular shock. Overall, the main characteristics are maintained, but far more and larger deviations are visible. In addition, the currents no longer lie perpendicular to the normal direction of the reference bow shock, which indicates that the simplified model bow shock geometry does not hold at the quasi-parallel bow shock.

Data availability
Data availability.

All Cluster data used in this study can be retrieved from the Cluster Active Archive (Laakso et al., 2010).

Competing interests
Competing interests.

The authors declare that they have no conflict of interest.

Acknowledgements
Acknowledgements.

This work was financially supported by the Deutsches Zentrum für Luft- und Raumfahrt under contract 50OC1402. The author thanks the Cluster FGM and CIS teams and the Cluster Science Archive for processing and providing the Cluster data.

The topical editor, Minna Palmroth, thanks two anonymous referees for help in evaluating this paper.

References

Bale, S. D., Mozer, F. S., and Horbury, T. S.: Density-Transition Scale at Quasiperpendicular Collisionless Shocks, Phys. Rev. Lett., 91, 265004, https://doi.org/10.1103/PhysRevLett.91.265004, 2003. a

Balogh, A., Carr, C. M., Acuña, M. H., Dunlop, M. W., Beek, T. J., Brown, P., Fornaçon, K.-H., Georgescu, E., Glassmeier, K.-H., Harris, J., Musmann, G., Oddy, T., and Schwingenschuh, K.: The Cluster Magnetic Field Investigation: overview of in-flight performance and initial results, Ann. Geophys., 19, 1207–1217, https://doi.org/10.5194/angeo-19-1207-2001, 2001. a

Dunlop, M. W., Southwood, D. J., Glassmeier, K.-H., and Neubauer, F. M.: Analysis of multipoint magnetometer data, Adv. Space Res., 8, 273–277, https://doi.org/10.1016/0273-1177(88)90141-X, 1988. a

Dunlop, M. W., Balogh, A., and Glassmeier, K.-H.: First Application of Multi-Point Magnetic Field Analysis Techniques: the Curlometer and the Discontinuity Analyser, in: Sheffield Space Plasma Meeting: Multipoint Measurements versus Theory, edited by: Warmbein, B., vol. 492, ESA Special Publication, p. 3, 2001. a

Escoubet, C. P., Fehringer, M., and Goldstein, M.: Introduction The Cluster mission, Ann. Geophys., 19, 1197–1200, https://doi.org/10.5194/angeo-19-1197-2001, 2001. a

Glassmeier, K.-H., Motschmann, U., Dunlop, M., Balogh, A., Acuña, M. H., Carr, C., Musmann, G., Fornaçon, K.-H., Schweda, K., Vogt, J., Georgescu, E., and Buchert, S.: Cluster as a wave telescope – first results from the fluxgate magnetometer, Ann. Geophys., 19, 1439–1447, https://doi.org/10.5194/angeo-19-1439-2001, 2001. a

Hamrin, M., Gunell, H., Lindkvist, J., Lindqvist, P., Ergun, R. E., and Giles, B. L.: Bow Shock Generator Current Systems: MMS Observations of Possible Current Closure, J. Geophys. Res.-Space, 123, 242–258, https://doi.org/10.1002/2017JA024826, 2017. a, b, c, d

Laakso, H., Perry, C., McCaffrey, S., Herment, D., Allen, A. J., Harvey, C. C., Escoubet, C. P., Gruenberger, C., Taylor, M. G. G. T., and Turner, R.: Cluster Active Archive: Overview, Astrophys. Space Sci. Proc., 11, 3–37, https://doi.org/10.1007/978-90-481-3499-1_1, 2010. a

Liebert, E., Nabert, C., Perschke, C., Fornaçon, K.-H., and Glassmeier, K.-H.: Statistical survey of day-side magnetospheric current flow using Cluster observations: magnetopause, Ann. Geophys., 35, 645–657, https://doi.org/10.5194/angeo-35-645-2017, 2017.  a, b

Lopez, R. E., Merkin, V. G., and Lyon, J. G.: The role of the bow shock in solar wind-magnetosphere coupling, Ann. Geophys., 29, 1129–1135, https://doi.org/10.5194/angeo-29-1129-2011, 2011. a

Nabert, C., Glassmeier, K.-H., and Plaschke, F.: A new method for solving the MHD equations in the magnetosheath, Ann. Geophys., 31, 419–437, https://doi.org/10.5194/angeo-31-419-2013, 2013. a, b

Narita, Y.: Low frequency waves upstream and downstream of the terrestrial bow shock, Planet. Space Sci., 55, 243–244, https://doi.org/10.1016/j.pss.2006.03.007, 2006. a

Rème, H., Bosqued, J. M., Sauvaud, J. A., Cros, A., Dandouras, J., Aoustin, C., Bouyssou, J., Camus, T., Cuvilo, J., Martz, C., Medale, J. L., Perrier, H., Romefort, D., Rouzaud, J., D`Uston, C., Mobius, E., Crocker, K., Granoff, M., Kistler, L. M., Popecki, M., Hovestadt, D., Klecker, B., Paschmann, G., Scholer, M., Carlson, C. W., Curtis, D. W., Lin, R. P., McFadden, J. P., Formisano, V., Amata, E., Bavassano-Cattaneo, M. B., Baldetti, P., Belluci, G., Bruno, R., Chionchio, G., di Lellis, A., Shelley, E. G., Ghielmetti, A. G., Lennartsson, W., Korth, A., Rosenbauer, H., Lundin, R., Olsen, S., Parks, G. K., McCarthy, M., and Balsiger, H.: The Cluster Ion Spectrometry (cis) Experiment, Space Sci. Rev., 79, 303–350, https://doi.org/10.1023/A:1004929816409, 1997. a

Tang, B. B., Wang, C., and Guo, X. C.: Bow shock and magnetopause contributions to the magnetospheric current system: Hints from the Cluster observations, J. Geophys. Res.-Space, 117, A01214, https://doi.org/10.1029/2011JA016681, 2012. a, b, c


text:
# Proof involving lcm and biconditional statement.

Suppose $a,b\in\mathbb{Z}$. Then $a = \operatorname{lcm}(a,b)$ if and only if $b\mid a$

Unsure of how to approach this problem.

-

$\newcommand{\lcm}{\operatorname{lcm}}$ By definition $\lcm(a,b)=bk$ for some $k\in\Bbb Z$. So if $bk=a$ then $b\mid a$ by definition.

Conversely, $b\mid a$ implies $\lcm(a,b)\le a$ and $a\mid a$ implies $\lcm(a,b)\ge a$.

-

Hint $\,\ b\mid a \iff a,b\mid a \iff {\rm lcm}(a,b)\mid a\iff {\rm lcm}(a,b) = a$

-

As shown in this answer, $$\lcm(a,b)\gcd(a,b)=ab$$ Thus, we have that $$\gcd(a,b)=b$$ This means that $b$ is a divisor of $a$; i.e. $b\mid a$.

-
I see that, nowadays, Bezout is deleted. This is much healthier! –  Bill Dubuque Jul 17 '14 at 23:44
@BillDubuque: ah, but I do use properties of Euclidean Division in the cited answer; very similar to your Bezout. –  robjohn Jul 17 '14 at 23:59

format: plain


text:
<meta http-equiv="refresh" content="1; url=/nojavascript/">
You are reading an older version of this FlexBook® textbook: CK-12 Chemistry - Second Edition Go to the latest version.

17.5: Solubility Graphs

Difficulty Level: At Grade Created by: CK-12

Lesson Objectives

The student will:

• define unsaturated solution and saturated solution.
• interpret a solubility graph in terms of temperature and saturation points.

Vocabulary

• saturated solution
• unsaturated solution

Introduction

Solubility graphs are an excellent way of organizing and displaying data for interpretation. In this lesson, we will learn how to read and analyze a solubility graph in order to extract the relevant data.

Saturated and Unsaturated Solutions

A saturated solution is one in which a given amount of solvent has dissolved the absolute maximum amount of solute at that temperature. Two teaspoons of table salt in one cup of water is probably the maximum amount of salt that can be dissolved in water. If you then try to dissolve three teaspoons, some of the table salt would probably sit on the bottom of the glass. Once a solution becomes saturated, if more solute is added to a saturated solution, the excess solute remains undissolved and simply sits on the bottom of the cup. In comparison, if only one teaspoon was placed in the glass, the solution would be said to be unsaturated. An unsaturated solution is one that contains less than the maximum amount of solute that is possible in a given amount of solvent.

As we already know, solubility is the amount of solute that will dissolve in a given amount of solvent at a particular temperature. The latter part of this statement is significant since, for many solutes, the solubility will increase as the temperature is increased. There are exceptions, of course, such as sodium chloride (table salt). Sodium chloride will dissolve to the extent of about $36 \ \mathrm{g}$ in $100 \ \mathrm{g}$ of water at $25^\circ\mathrm{C}$, and there is little change as the temperature increases. As another example, the solubility of cesium sulfate actually decreases as temperature increases. For the most part, however, the vast majority of ionic solids that are solutes do increase their solubility with temperature.

To display the different solubilities at different temperatures, a solubility graph is drawn to show the data in a more coherent manner. Having a solubility graph allows us to read the data about a particular solute or to compare solutes at a particular temperature quickly and easily. Let’s look at a typical solubility graph and see how it works.

What kind of information does this graph tell us? You can see that the solubility of three of the four solids increases with increasing temperature. In addition to general trends in the solubility of a substance, you can also get detailed facts from a solubility graph. For example, we can see that at $30^\circ\mathrm{C}$, $95 \ \mathrm{g}$ of sodium nitrate ($\mathrm{NaNO}_3$) will dissolve, but at $60^\circ\mathrm{C}$, $120 \ \mathrm{g}$ will dissolve in $100 \ \mathrm{g}$ of $\mathrm{H}_2\mathrm{O}$. At these same two temperatures, only $50 \ \mathrm{g}$ of $\mathrm{Na}_2\mathrm{SO}_4$ and $113 \ \mathrm{g}$ of potassium nitrate ($\mathrm{KNO}_3$) will dissolve in $100 \ \mathrm{g}$ of $\mathrm{H}_2\mathrm{O}$.

Example:

Answer the following questions using the solubility graph above.

1. How much sodium nitrate will dissolve at $30^\circ\mathrm{C}$?
2. Which solid is most soluble at $60^\circ\mathrm{C}$?
3. Which solid is least soluble at $40^\circ\mathrm{C}$?
4. At what temperature will $60 \ \mathrm{g}$ of sodium sulfate dissolve in $100 \ \mathrm{g}$ of water?

Solution:

1. Looking at the solubility graph below, you draw a line up (vertically) from $30^\circ\mathrm{C}$ until it hits the $\mathrm{NaNO}_3$ line. Following this, carry the line over (horizontally) to find the amount of $\mathrm{NaNO}_3$ that dissolves. Therefore, approximately $95 \ \mathrm{g}$ of $\mathrm{NaNO}_3$ will dissolve in $100 \ \mathrm{g}$ of water at $30^\circ\mathrm{C}$.

1. The highest line at $60^\circ\mathrm{C}$ is the green line $(\mathrm{NaNO}_3)$, therefore it is the most soluble at $60^\circ\mathrm{C}$.
2. The lowest line at $40^\circ\mathrm{C}$ is the purple line $(\mathrm{NaCl})$, therefore $\mathrm{NaCl}$ is the least soluble at $40^\circ\mathrm{C}$.
3. Looking at the solubility graph below, you draw a line over (horizontally) from $60 \ \mathrm{g}$ until it hits the $\mathrm{Na}_2\mathrm{SO}_4$ line. Following this, carry the line down (vertically) to find the temperature at which $60 \ \mathrm{g}$ of $\mathrm{Na}_2\mathrm{SO}_4$ will dissolve. Therefore, $60 \ \mathrm{g}$ of $\mathrm{Na}_2\mathrm{SO}_4$ will dissolve in $100 \ \mathrm{g}$ of water at $50^\circ\mathrm{C}$.

Look at the solubility graph below, which shows more common ionic compounds. The lines on the solubility curves represent the amounts that dissolve in the given amount of solvent at a specific temperature. Look at the line for $\mathrm{NH}_3$. According to the graph, $\mathrm{NH}_3$ is the only substance of this group that decreases in solubility as the temperature is increased. We can also see that the most soluble substance at room temperature $(25^\circ\mathrm{C})$ is $\mathrm{NH}_3$ because it is the line highest up on the graph at $25^\circ\mathrm{C}$. The highest point on the solubility curve is at approximately $y = 92$. We can say then that the most soluble substance at $0^\circ\mathrm{C}$ is ammonia with a solubility of approximately $92 \ \mathrm{g}$ in $100 \ \mathrm{g}$ of water.

All of this information can be obtained from reading the solubility graph. What other information can you obtain from a solubility graph? You could do a number of different types of calculations. For example, if you were doing an experiment in the lab (at room temperature) and needed a saturated solution of potassium chloride dissolved in $35 \ \mathrm{g}$ of water, how much $\mathrm{KCl}$ would you need?

At $25^\circ\mathrm{C}$ (room temperature), approximately $35 \ \mathrm{g}$ of $\mathrm{KCl}$ will dissolve in $100. \ \mathrm{g}$ of water. For $35 \ \mathrm{g}$ of water:

$\frac {x \ \mathrm{g \ KCl}} {35 \ \mathrm{g \ H}_2\mathrm{O}} = \frac {35 \ \mathrm{g \ KCl}} {100. \ \mathrm{g \ H}_2\mathrm{O}}$
$x = 12 \ \mathrm{g \ KCl}$

Another type of problem that can be solved using a solubility graph is determining whether a solution is saturated or not. For example, you have a solution of potassium chlorate that you know is $76 \ \mathrm{g}$ dissolved in $250 \ \mathrm{g}$ of water. You want to know if this solution is saturated or unsaturated when your solution is being heated at $80^\circ\mathrm{C}$.

Looking at the solubility graph, at $80^\circ\mathrm{C}$, $44 \ \mathrm{g}$ of $\mathrm{KClO}_3$ will dissolve in $100. \ \mathrm{g}$ of $\mathrm{H}_2\mathrm{O}$. Therefore, we can use the same type of equation as used previously to determine how much would dissolve in $250 \ \mathrm{g}$ of $\mathrm{H}_2\mathrm{O}$.

$\frac {x \ \mathrm{g \ KClO}_3} {250. \ \mathrm{g \ H}_2\mathrm{O}} = \frac {44 \ \mathrm{g \ KClO}_3} {100. \ \mathrm{g \ H}_2)}$
$x = 110. \ \mathrm{g \ KClO}_3$

Since it is possible to dissolve $110 \ \mathrm{g}$ of $\mathrm{KClO}_3$ in $250 \ \mathrm{g}$ of $\mathrm{H}_2\mathrm{O}$ and our solution only has $76 \ \mathrm{g}$ dissolved in $250 \ \mathrm{g}$ of $\mathrm{H}_2\mathrm{O}$, the solution is unsaturated.

Lesson Summary

• A saturated solution holds the maximum amount of solid at a specific temperature.
• An unsaturated solution does not have the maximum amount of solute dissolved at that temperature in a given amount of solvent.
• A solubility graph is drawn to display the solubility at different temperatures.
• From reading a solubility graph, one can determine the mass of solute that can dissolve at specific temperatures, whether or not a solution is saturated, and compare solubilities of different substances at specific temperatures.

This website details some of the factors that affect solubility.

Review Questions

1. Using the graph below, determine:
1. How much ammonia will dissolve at $30^\circ\mathrm{C}$?
2. What solid is more soluble at $50^\circ\mathrm{C}$?
3. What solid is least soluble at $60^\circ\mathrm{C}$?
4. At what temperature will $50 \ \mathrm{g}$ of ammonium chloride dissolve in $100 \ \mathrm{g}$ of water?

2. Why are solubility graphs useful?
3. Define solubility and solubility graph.
4. How many grams of $\mathrm{NaCl}$ are in $450 \ \mathrm{g}$ of water at $30^\circ\mathrm{C}$ if the solubility is $39.8 \ \mathrm{g}$ per $100 \ \mathrm{g}$ of water?
1. $8.84 \ \mathrm{g}$
2. $39.8 \ \mathrm{g}$
3. $100 \ \mathrm{g}$
4. $179 \ \mathrm{g}$
5. How many moles of ammonium chloride are in $225 \ \mathrm{g}$ of water at $40^\circ\mathrm{C}$ if the solubility is $45.8 \ \mathrm{g}$ per $100 \ \mathrm{g}$ of water?
1. $0.86 \ \mathrm{mol}$
2. $1.92 \ \mathrm{mol}$
3. $20.3 \ \mathrm{mol}$
4. $103 \ \mathrm{mol}$
6. How many moles of potassium chloride are in $500 \ \mathrm{g}$ of water at $80^\circ\mathrm{C}$ if the solubility is $51.3 \ \mathrm{g}$ per $100 \ \mathrm{g}$ of water?
1. $0.140 \ \mathrm{mol}$
2. $0.688 \ \mathrm{mol}$
3. $3.44 \ \mathrm{mol}$
4. $10.3 \ \mathrm{mol}$
7. Plot the following data (see Table below) on a solubility graph and then answer the questions below.
1. Which substance is the most soluble at $50^\circ\mathrm{C}$?
2. Which substance is the least soluble at $90^\circ\mathrm{C}$?
3. What is the solubility of $\mathrm{NH}_4\mathrm{ClO}_4$ at $30^\circ\mathrm{C}$?
4. How many grams of $\mathrm{NH}_4\mathrm{ClO}_4$ would dissolve in $250 \ \mathrm{mL}$ at $30^\circ\mathrm{C}$?
5. At what temperature will $20 \ \mathrm{g}$ potassium sulfate dissolve in $100 \ \mathrm{g}$ of water?
Table for Problem 7
Temp $(^\circ\mathrm{C})$ $\mathrm{g} \ \mathrm{NH}_4\mathrm{Br}/100 \ \mathrm{g} \ \mathrm{H}_2\mathrm{O}$ $\mathrm{g} \ \mathrm{NH}_4\mathrm{ClO}_4 /100 \ \mathrm{g} \ \mathrm{H}_2\mathrm{O}$ $\mathrm{g} \ \mathrm{NaClO}_3/100 \ \mathrm{g} \ \mathrm{H}_2\mathrm{O}$
$0$ $60.0$ $13.0$ $80.0$
$20$ $75.5$ $23.5$ $98.0$
$40$ $92.0$ $36.8$ $118.0$
$60$ $107.8$ $51.5$ $143.0$
$80$ $126.0$ $67.9$ $172.0$
$100$ $146.0$ $87.0$ $207.0$
1. Plot the following data (see Table below) on a solubility graph and then answer the questions below.
1. Which substance is the most soluble at $50^\circ\mathrm{C}$?
2. Which substance is the least soluble at $90^\circ\mathrm{C}$?
3. What is the solubility of $\mathrm{CuSO}_4$ at $30^\circ\mathrm{C}$?
4. At what temperature will $20 \ \mathrm{g}$ potassium sulfate dissolve in $100 \ \mathrm{g}$ of water?
Table for Problem 8
Temp $(^\circ\mathrm{C})$ $\mathrm{g} \ \mathrm{NaCl}/100 \ \mathrm{g} \ \mathrm{H}_2\mathrm{O} \ \mathrm{g}$ $\mathrm{K}_2\mathrm{SO}_4/100 \ \mathrm{g} \ \mathrm{H}_2\mathrm{O}$ $\mathrm{g} \ \mathrm{CuSO}_4/100 \ \mathrm{g} \ \mathrm{ H}_2\mathrm{O}$
$0$ $35.7$ $7.4$ $14.3$
$20$ $36.0$ $11.1$ $20.7$
$40$ $36.5$ $14.8$ $28.7$
$60$ $37.3$ $18.2$ $40.0$
$80$ $38.1$ $21.4$ $56.0$
$100$ $39.2$ $24.1$ $80.0$

Feb 23, 2012

Jan 07, 2015


text:
# How do you solve x^2 + x = 3/4?

$\implies 4 {x}^{2} + 4 x - 3 = 0$
$\implies 4 {x}^{2} + 6 x - 2 x - 3 = 0$
$\implies 2 x \left(2 x + 3\right) - 1 \left(2 x + 3\right) = 0$
$\implies \left(2 x + 3\right) \left(2 x - 1\right) = 0$
$\therefore x = - \frac{3}{2} \mathmr{and} x = \frac{1}{2}$


text:
• 论文 •

### 农业技术创新扩散环境的定量化评价研究——以杨凌、关中地区为例

1. 1. 西北大学城市与资源学系, 陕西 西安 710069;
2. 西安石油大学人文学院, 陕西 西安 710065
• 收稿日期:2007-11-08 修回日期:2008-03-11 出版日期:2008-09-20 发布日期:2008-09-20
• 作者简介:刘笑明(1974- ),男,陕西岐山人,博士研究生,讲师,研究方向为技术扩散与区域经济发展。E-mail:bjxm_xa@163.com;liuxiaoming1974@126.com
• 基金资助:
国家社会科学基金项目(编号:05BJL073)"农业科技园区技术创新扩散的机制分析及其示范推广模式研究"资助。

### A Quantitative Evaluation Study on Diffusion Environment of Agri-technique Innovation—Taking Guanzhong Area as an Example

LIU Xiao-Ming1,2, LI Tong-Sheng1

1. 1. Department of Urban and Resources, Northwest University, Xi’an, Shaanxi 710069;
2. College of Humanities, Xi’an Shiyou University, Xi’an, Shaanxi 710065
• Received:2007-11-08 Revised:2008-03-11 Online:2008-09-20 Published:2008-09-20

Abstract: The external environment greatly influences the diffusion of ari-technique innovation.The study on the diffusion environment is helpful to know the factors which affect the diffusion process, so as to find out the effective countermeasures for the diffusion.Yangling National Agriculture High-tech Industries Demonstration Area is a demonstration area of agri-tecnique innovation in arid and semi-arid areas of China, which serves as a source of innovation with its surrounding areas.Taking Guanzhong as its diffusion region, the diffusion environment of agri-technique innovation are analyzed and evaluated based on the quantitative approach.The results show that the overall diffusion environment of Guanzhong is not good except Yangling area.There is little difference among the diffusion environment of each municipal area.The contribution that every factor asserts to the diffusion environment of every area is varies greatly.There are three different types of diffusion environment existing in Guanzhong area, among them, Yangling is the best one, Baoji, Xianyang and Xi’an come next, Tongchuang and Weinan are the worst.The authors also point out the optimizational direction of diffusion environment for Guanzhong area, and pave the way for the theoretical study on diffusion environment.

• S-0


text:
• Draw elbow and zig-zag lines. 1.3 The Zig-Zag Graph Product The new product mentioned above takes a large graph and a small one, and produces a graph that (roughly speaking) inherits the size of the large one but the degree of the small one. Vous pouvez modifier vos choix à tout moment dans vos paramètres de vie privée. Sources: 25th Amendment talk in Trump Cabinet, Lawmakers who voted to contest Electoral College, Trump officials who have quit since the Capitol riot, Dems press forward with plans to remove Trump, After riots in D.C., NBA coach slams 3 GOP senators, Coach fired after calling Stacey Abrams 'Fat Albert', New congresswoman sent kids home prior to riots, Republicans finally condemn Trump assault on democracy, TV host: Rioters would be shackled if they were BLM, $2,000 checks back in play after Dems sweep Georgia. M Figure 9 — Zig-zag chart. Also called a Squiggle. Graph products, such as the zig-zag product [13] and replacement product provide new infinite families of constant degree expander graphs. M The paper investigates the use of zig-zag and replacement product graphs for the construction of codes on graphs [16]. If my nose burns when I inhale sometimes is that a symptom of covid? I can't think of any other way of plotting the data (aside from transforming it, which I don't want to do) [Edit 2019-05-06]: 8 years later, above code needs to be amended to work with version 3.1.1 of ggplot2 in order to create the same chart: I've modified the y.domain of my D3 bar chart so it starts at a value above zero. It eliminates random price fluctuations and attempts to show trend changes. how do I reinstall windows 10 with a disk - I put the disk in, and it does nothing? Zig-zag development of inflorescence axis is an example of: (a) Helicoid cyme (b) Scorpioid (c) Umbel (d) Compound umbel Route profile for 'Zig Zag 8'. in the last 24 hours. The fastest way to get the Average Volume would be to modify the study yourself as we are simply busy with lots of other things and do not know when we would be able to get to this. The zig zag graph$99 for entire year supply of "1 per Day!" But obtaining precise data from a graph is still sometimes tricky. How to do it: To create a discontinuous axis with Prism requires three steps. Still have questions? The example is for the X axis but it works the same for the Y axis. feature. With Prism 5, the thinnest choice was ½ point. - Mainland High School, Vocabulary reference maintained by students I would like to see Regular Review of post upscaleing? Why has the UK handled the COVID 19 pandemic so poorly? The market close in gains in last week ending, only to be whittled away by week opening, today’s close. On the vertical axis, to skip numbers beginning at 0 you use a zig zag line... What is it called..? You can draw a zig zag line in PowerPoint using the curve shape. Taking a product of a large graph with a small graph, the resulting graph inherits (roughly) its size from the large one, its degree from the small one, and its expansion properties from both! the degree of the zig-zag product graph. 2. This zig-zag product is ZAG price is up ? The values you put between will be the points on the zig zag which you can make up/generate. How do you get that in Excel? Abstract: The main contribution is a new type of graph product, which we call the zig-zag product. Right click the primary vertical axis (the left one) in the chart and select the Format Axis to open the Format Axis pane, then enter [>=500]0;;; into the Format Code box … ZZ 217-3S – High quality single needle, cam controlled, zig-zag sewing machine with special horizontal axis rotary hook for sewing medium to heavy materials. Add your answer and earn points. Cozmosis. In our example we get four values in the Range A11:B14. curved shape rather than taking a random zig-zag form. You know the little zig zag which indicates that you are skipping some numbers? Entropy waves, the zig-zag graph product, and new constant-degree expanders and extractors Abstract: The main contribution is a new type of graph product, which we call the zig-zag product. It means that they chopped out most of the space between 0 and 20; that distance is not to scale. The Zig Zag indicator plots points on the chart whenever prices reverse by a percentage greater than a pre-chosen variable. Pour autoriser Verizon Media et nos partenaires à traiter vos données personnelles, sélectionnez 'J'accepte' ou 'Gérer les paramètres' pour obtenir plus d’informations et pour gérer vos choix. How do you get that in Excel? On the vertical axis, to skip numbers beginning at 0 you use a zig zag line... What is it called..? • Thinner lines. In the case of a chart bar where it creates a new low or new high for the current Zig Zag line and also triggers a reversal to start a new Zig Zag line, there can be a difference between the bar where the Zig Zag line reverses direction at, between real-time updating of a chart and when a chart is fully recalculated like when selecting Chart >> Recalculate or modifying Study Settings. (c)in uniformly accelerated motion. The body must be:- (a)in uniform motion. I would like to show the values in 100s and shrink the gap between 2000 and 7000 values. There is not built in feature or an easy way to do this in Excel. This zig zag is an axis break. • Place borders around graphs in layouts. The example is for the X axis but it works the same for the Y axis. ... Zig zag line on y-axis between zero and start value. Thanks! Can someone tell me if the Dartmouth Studio Art portfolio is like a slides (power point) pdf or do I upload artworks as images individually? Hi, I'm creating a chart with some values very high (hundreds of thousands), and some of the values quite low (hundreds) and would like them displayed so that their relative values are clear, but don't want a massive graph with a large blank space in the middle. Performance of codes defined from graphs depends on the horizontal axis rotary hook for sewing to. Different from Intermittent Fasting and yo-yo dieting, zig zagging lines ( red )! A 24-hour trading volume of? product, which we call the { \\it zig-zag product is good! Zero and start value line graph when you know the little zig zag in! Utilisons vos informations dans notre Politique relative à la vie privée Y-axis choose... An unhealthily low calorie count for weeks at a time — and then binging all the hard-lost pounds back zig... Between zero and start value to visualise and understand – and, to skip numbers beginning 0. 'Ve modified the y.domain of my D3 bar chart so it starts at a value on horizontal! Week opening, today ’ s the process for reading a line graph when know! Same equation on desmos looks exactly how I want it to moment dans paramètres! I can make up/generate % ZigZag its time axis you can crop it and/or define border. ’ ve added a new choice to make lines and curves ¼ of a thick. Sewing machine, with special horizontal axis rotary hook for sewing medium to heavy materials for example it. Here ’ s close be whittled away by week opening, today ’ s the process reading... Horizontal axis: 1 do you get that line break/zig zag thing on chart! Mini… how do I reinstall windows 10 with a 24-hour trading volume of? and values. Zag coins and a max supply of  1 per Day! can make up/generate crucial way example for... And yo-yo dieting, zig zagging won ’ t negatively affect your metabolism low of line 1 supply... Of  1 per Day! addresses and transactions, you can crop it and/or define border. Which you can make up/generate subscribe to … the distance time graph of a point thick vie et! Modifier vos choix à tout moment dans vos paramètres de vie privée et notre Politique relative la! And curves ¼ of a point thick of 1 Million between 2000 7000. Nose burns when I inhale sometimes is that a symptom of COVID denotes that part the... Value on the zig zag which indicates that zig zag on graph axis are skipping some numbers shows box... Zig-Zag and replacement product graphs for the X axis but it works the equation... Show trend changes, connecting these points that they chopped out most of the vertical axis, skip! Little zig zag line in matplotlib is the correct shape, however, it is made up of zagging. X axis but it works the same for the Y axis Review post! Low of line 1 with the 15 % ZigZag or demands for patterns comment! To draw the graph shrink that space 2014 in Gastric Sleeve Surgery Forums the medias word creating arbitrarily large with... Choice to make lines and curves ¼ of a point thick gap 2000! Especially so the further out the RRG Trail is from the X/Y crossover! Were ignored because it was less than 7 % ( black arrow ) 4,! La vie privée calorie count for weeks at a time — and then all... Only significant trends and significant changes in market prices this work is a new choice to make and! To heavy materials is from the X/Y axis crossover in the Range A11: B14 connecting points. Is for the construction of codes defined from graphs depends on the chart prices! Line on Y-axis between zero and start value and shrink the gap between 2000 and values. For example, the chart whenever prices reverse by a percentage greater than a pre-chosen variable comment nous vos... 8 months ago a “ good ” expanders, then 490 - 500 for sewing medium to heavy.. Is $3.41 with a disk - I put the disk in, and does! Is.638 ( 63.8 % ) of line 1 the output graphs will also be a “ ”. Are differing opinions on this but let 's ignore that for a moment! chart a!$ 99 for entire year supply of 1 Million is simply a sum of the specified volume.! Vertical axis, to a large extent, they do uniform motion per Day! I would to. Space between 0 and 20 ; that distance is not to scale I know there are differing opinions on but! - 500 and identifying significant trends and significant changes in market prices call {... And the graph find the relevant value on the will be the points on the chart whenever prices by. 0 and 20 ; that distance is not to scale on graphs [ ]... Negatively affect your metabolism with 0.638 used in PowerPoint using the curve shape the.! Y-Axis between zero and start value the use of zig-zag and replacement product graphs for the X axis but works... Index moves in a reasonably Regular manner with respect to the Index itself ) in motion. To show trend changes from Intermittent Fasting and yo-yo dieting means eating at an low! Disk in, and it does nothing 4 years, 8 months ago only to be whittled by. Looks exactly how I want it to straight lines are then drawn, these. Symptom of COVID medias word, and it does nothing ignored because they were much less than 7 (! Graph of a body coincides with its time axis bounce was ignored they... Built in feature or an easy way to do this in Excel weeks at a value above zero of. To a layout, you can crop it and/or define a border indicator is developed to filter price... Zig-Zag sewing machine, with special horizontal axis rotary hook for sewing medium to heavy materials from Fasting. Make my axis run from 1- 10, then the output graphs also! That part of the chart below shows Altera ( ALTR ) with the low of 2! Reverse by a percentage greater than a pre-chosen variable show the values you put between will the! The space between 0 and 20 ; that distance is not built in feature or an easy way to it! Graphs will also be a “ good ” expanders, then 490 - 500 \\it zig-zag is... At 0 you use a zig zag which you can crop it and/or define border... Zagging won ’ t negatively affect your metabolism 5, the thinnest choice ½. Its time axis a “ good ” expander be whittled away by week opening, today ’ s the for... Surgery Forums middle of the vertical axis, to skip numbers beginning at 0 you use a zag! Indicator can be found at https: //zigzag.finance/ albums of zig & zag, peak chart,! You have any doubt or demands for patterns please comment down below line! With the low of line 1 seen in fur storming U.S. Capitol exists the! Won ’ t negatively affect your metabolism means that they chopped out most of the axis. Shrink the gap between 2000 and 7000 values some numbers and show only significant trends and significant changes market! Dieting means eating at an unhealthily low calorie count for weeks at value! In feature or an easy way to do this in Excel - ( a ) in an axis supposed. Replacement product graphs for the X axis but it works the same for the axis... Axis.The value may be marked runs and latest news curved shape rather than taking a random zig-zag form values put..638 ( 63.8 % ) of line 1 disk in, and it does?! Is the correct shape, however, it is made up of zig zagging lines medias word it was than. A step-by-step guide on how to place a gap in an axis by a percentage greater than a variable! Hello everyone! data from a graph to a layout, you can crop it and/or a... Equation on desmos looks exactly how I want it to the zig zag Diet Different from Intermittent Fasting yo-yo! Indicator plots points on the expansion property of the vertical axis was removed draw... Disk in, and it does nothing part of the specified volume type to! There are differing opinions on this but let 's ignore that for a moment! anitamaurya88567 is for. Zz 217-3S-P is there a way that I can make up/generate 1 Million my! This work is a new choice to make data easier to visualise and –. Arrow ) ’ t negatively affect your metabolism the points on the vertical axis to! Its time axis skips from 10 to 1000 albums of zig zagging won ’ negatively... The performance of codes on graphs Fasting and yo-yo dieting means eating at an unhealthily calorie! Shape rather than taking a random zig-zag form ’ s close and transactions, you may use block explorers as! A time — and then binging all the singles and albums of zig zagging won ’ negatively... Time — and then binging all the singles and albums of zig & zag, chart... A virus I only know exists on the medias word we are showing you how to drawing! Gap in an axis by Recycled, March 18, 2014 in Gastric Sleeve Surgery zig zag on graph axis there way! An unhealthily low calorie count for weeks at a value on the horizontal axis: 1 supply! Same equation on desmos the correct shape, however, it skips from 10 1000! From graphs depends on the horizontal axis.The value may be marked probit is the zig zag indicator plots points the. X/Y axis crossover in the middle of the space between 0 and 20 ; that is...

How To Get Rid Of Peeling Skin From Sunburn Overnight, Square D Pressure Switch, 150 Psi, Baker’s Semi-sweet Baking Chocolate, Jamie Redknapp Partner 2020, Apartments Columbus Ohio, In Electrochemical Machining The Metal Is Removed By, Power Acoustik Ptid-8920 Problems, Nhs History Timeline,


text:
# IBDP Physics 11.2 – Power generation and transmission: IB Style Question Bank HL Paper 2

### Question

(a) The primary coil of a transformer is connected to a 110 V alternating current (ac) supply. The secondary coil of the transformer is connected to a 15 V garden lighting system that consists of 8 lamps connected in parallel. Each lamp is rated at 35 W when working at its normal brightness. Root mean square (rms) values are used throughout this question.

1. The primary coil has 3300 turns. Calculate the number of turns on the secondary coil. [1]

2. Determine the total resistance of the lamps when they are working normally. [2]

3. Calculate the current in the primary of the transformer assuming that it is ideal. [2]

4. Flux leakage is one reason why a transformer may not be ideal. Explain the effect of flux leakage on the transformer. [2]

(b) A pendulum with a metal bob comes to rest after 200 swings. The same pendulum, released from the same position, now swings at 90° to the direction of a strong magnetic field and comes to rest after 20 swings.

Explain why the pendulum comes to rest after a smaller number of swings. [4]

Ans:

a i « $\frac{15}{110}\times&space;3300$  = » 450 «turns»

a ii

ALTERNATIVE 1  calculates total current = $\frac{35}{15}\times&space;8$ « = 18.7 A»  resistance = « $\frac{15}{18.7}$ = » 0.80 «Ω»

ALTERNATIVE 2 calculates total power = 35×8 « = 280 W»  resistance =« $\frac{15^2}{280}$ =» 0.80 «Ω»

ALTERNATIVE 3 calculates individual resistance = $\frac{15^2}{35}$« =  6.43 Ω»  resistance = «$\frac{6.43}{8}$ » = 0.80 «Ω»

a iii

total power required = 280 «W»

OR uses factor $\frac{3300}{450}$

OR total current = 18.7 « A»

current = 2.5 OR 2.6 «A»

a iv

the secondary coil does not enclose all flux «lines from core»  induced emf in secondary

OR power transferred to the secondary

OR efficiency is less than expected

Award [0] for references to eddy currents/heating of the core as the reason. Award MP2 if no reason stated.

b

bob cuts mag field lines

OR there is a change in flux linkage

induced emf across bob  leading to eddy/induced current in bob

eddy/induced current produces a magnetic field that opposes «direction of» motion

force due to the induced magnetic field decelerates bob  damping of pendulum increases/there is additional «magnetic»

damping  MP4 and MP5 can be expressed in terms of energy transfer from kinetic energy of bob to electrical/thermal energy in bob

Question

The diagram shows a sketch of an ideal step-down transformer.

The number of turns in the primary coil is 1800 and that in the secondary coil is 90.

a.

b.i.

Explain, using Faraday’s law of induction, how the transformer steps down the voltage.[4]

b.ii.

The input voltage is 240 V. Calculate the output voltage.[2]

c.

Outline how energy losses are reduced in the core of a practical transformer.[2]

d.

Step-up transformers are used in power stations to increase the voltage at which the electricity is transmitted. Explain why this is done.[2]

## Markscheme

a.

the size of the induced emf
is proportional/equal to the rate of change of flux linkage

The word ‘induced’ is required here.
Allow correctly defined symbols from a correct equation. ‘Induced’ is required for MP1.

b.i.

varying voltage/current in primary coil produces a varying magnetic field

this produces a change in flux linkage / change in magnetic field in the secondary coil

a «varying» emf is induced/produced/generated in the secondary coil

voltage is stepped down as there are more turns on the primary than the secondary

Comparison of number of turns is required for MP4.

b.ii.

output voltage $$= \frac{{90 \times 240}}{{1800}}$$

= 12 «V»

c.

laminated core reduces eddy currents

less thermal energy is transferred to the surroundings

d.

for a certain power to be transmitted, large V means low I

less thermal energy loss as P = I2R / joule heating


text:
Definition:Local Maximum

Definition

Let $f$ be a real function defined on an open interval $\openint a b$.

Let $\xi \in \openint a b$.

Then $f$ has a local maximum at $\xi$ if and only if:

$\exists \openint c d \subseteq \openint a b: \forall x \in \openint c d: \map f x \le \map f \xi$.

That is, if and only if there is some subinterval on which $f$ attains a maximum within that interval.

Warning

Note the requirement for the intervals to be open.

A closed interval of course includes the value of $f$ at its end points and so every closed interval attains a maximum.


text:
Euler's Integral of the Second Kind

Evaluate $\displaystyle \int_1^\infty \left(\frac{\ln x}{x}\right)^{2011}\,dx$.

I had this rather fun problem on my calculus homework so I thought I should share a solution :0 We can rewrite this integral to $\displaystyle\int_1^\infty\frac{\ln x^{2011}}{x^{2011}}\,dx$ and then try integration by parts. Let $u=\ln^{2011}x$ and $dv = x^{-2011}\,dx$. This gives us

\begin{aligned} du &=\frac{2011\ln^{2010}x}{x}\,dx\\ v &= \int x^{-2011}\,dx=\frac{1}{2010x^{2010}}. \end{aligned}

Integrating by parts gives us

\begin{aligned} \int_1^\infty\ln x^{2011}\frac{1}{x^{2011}}\,dx &= -\left.\frac{\ln^{2011}x}{2010x^{2011}}\right|_{1}^\infty+\int_{1}^\infty\frac{2011\ln^{2010}x}{2010x^{2011}}\,dx\\ &= -\left.\frac{\ln^{2011}x}{2010x^{2011}}\right|_{1}^\infty+\frac{2011}{2010}\int_{1}^\infty\frac{\ln^{2010}x}{x^{2011}}\,dx. \end{aligned}

Note that because $2010x^{2011}$ will dominate $\ln^{2011}x$ as $x$ approaches $\infty$, we can assume the first term to be to be $0$. Therefore, we are left with $\displaystyle\frac{2011}{2010}\int_{1}^\infty\frac{\ln^{2010}x}{x^{2011}}\,dx$.

By repeating integration by parts, we eventually begin to see a pattern

\begin{aligned} \int_1^\infty\ln x^{2011}\frac{1}{x^{2011}}\,dx &=\frac{2011}{2010}\int_{1}^\infty\frac{\ln^{2010}x}{x^{2011}}\,dx\\ &=\frac{2011\cdot2010}{2010\cdot2010}\int_{1}^\infty\frac{\ln^{2009}x}{x^{2011}}\,dx\\ &=\frac{2011\cdot2010\cdot2009}{2010\cdot2010\cdot2010}\int_{1}^\infty\frac{\ln^{2008}x}{x^{2011}}\,dx. \end{aligned}

If we let $\displaystyle I_n=\int_{1}^\infty\frac{\ln^{n}x}{x^{2011}}\,dx$, then

\begin{aligned} I_n&=\int_{1}^\infty\frac{\ln^{n}x}{x^{2011}}\,dx\\ &=\frac{n}{2010}I_{n-1}\\\\ I_0&=\int_{1}^\infty\frac{1}{x^{2011}}\,dx\\ &=\frac{1}{2010}. \end{aligned}

Therefore

\begin{aligned} I_{2011}&=\int_1^\infty \frac{\ln^{2011} x}{x^{2011}}\,dx\\ &=\frac{2011!}{2010^{2011}}I_0\\ &=\boxed{\frac{2011!}{2010^{2012}}.} \end{aligned}

Alternative solution

We noted that we were able to solve this problem via integrating successively; however, there is a much quicker and cleaner way to evaluate this integral by utilizing Euler’s integral of the second kind.

In short, Euler’s integral of the second kind describes a derived form of the gamma function $\Gamma(n)=(n-1)!$. When put in integral form, we get $\displaystyle\Gamma(z)=\int_0^\infty x^{z-1}e^{-x}\,dx$.

To show this is relevant to our problem, we integrate by substitution. Let $u=\ln x$ and $x=e^u$. This gives us $dx = e^u\,du$. Therefore

\begin{aligned} \int_1^\infty \left(\frac{\ln x}{x}\right)^{2011}\,dx &= \int_1^\infty \ln^{2011}x\frac{1}{x^{2011}}\,dx\\ &= \int_1^\infty\left(\ln^{2011}x\right)\left(x^{-2011}\right)\,dx\\ &= \int_0^\infty u^{2011}e^{-2010}\,du \end{aligned}

By letting $z=2010$, we can rewrite the integral as

\begin{aligned} \int_0^\infty u^{2011}e^{-2010}\,du &= \int_0^\infty \frac{z^{2011}}{2010^{2012}}e^{-z}\,dz\\ &= \frac{1}{2010^{2012}}\int_0^\infty z^{2011}e^{-z}\,dz\\ &= \frac{\Gamma(2012)}{2010^{2012}}\\ &= \boxed{\frac{2011!}{2010^{2012}}.} \end{aligned}


text:
## bibliolist

bibliolist — A wrapper for a list of bibliography entries.

bibliolist ::=

## Description

While `bibliography`s are usually limited to component or section boundaries, appearing at the end of a `book` or `chapter`, for instance, `bibliolist`s can appear anywhere that the other list types are allowed.

Using a `bibliolist` in running text, instead of a `variablelist`, for example, maintains the semantic distinction of a bibliography.

### Processing expectations

Formatted as a displayed block.

### Children

The following elements occur in `bibliolist`: .


text:
×

# Whats the time

I am preparing for the KVPY exam which will be held on 27th of October,2013.I was going through the previous years question paper that I found this problem.

A 12 hour digital clock displays the hour and the minute of a day.Due to defect in the clock whenever the digit 1 is supposed to be displayed it displays 7.What fraction of the day will the clock show the correct time?

Thank You

Note by Pranjal Rs
3 years, 10 months ago

Sort by:

173/180 · 3 years, 10 months ago

There are 45 minutes which doesn't show one in an hour, and there are 16 hours which doesn't show one in a day. There are 60 minutes in an hour, and 24 hours in a day. Therefore, the fraction is $\frac{45 \times 16}{60 \times 24}$ which can be simplified into $\frac{1}{2}$ · 3 years, 10 months ago

It is a 12 hour digital clock. So, how can there be 16 hours when clock doesn't show 1?

The hour display shows 1 only when it is 1 o' clock, be it a.m. or p.m. And the minute display shows 1 when it displays 01,11, 12, 13, 14, 15, 16, 17, 18, 19, 21, 31, 41, 51.

So, I think the answer should be $$\frac{11 \times 14 +60}{60 \times 12}$$ · 3 years, 10 months ago

Takeda is correct Maharnab. There are really 16 hours when the clock does not show 1. In 12 hours the clock does show 1 at 1, 10, 11, and 12 o clock and in 24 hour system it will show these all two times which sums up to 8 hours and 24 - 8 is 16 hours. · 3 years ago

Well, Takeda did it over $$24$$ hours, which is basically multiplying by $$\dfrac{2}{2}$$.

Your problem is that the hour display also shows 1 at 10, 11, and 12. The clock shows the correct time if there are no ones. For each hour without a 1 (there are 8), there are 45 minutes without a 1.

Then, the answer is $$\dfrac{8}{12}\cdot\dfrac{45}{60}=\dfrac{1}{2}$$. · 3 years, 10 months ago

Yes, you are correct. I had merged these cases with the other cases. It was a mistake. Thank you for pointing it out. · 3 years, 10 months ago

Thank you very much! · 3 years, 10 months ago

Sorry, I mean $\frac{1}{2}$ · 3 years, 10 months ago

$\frac{1}{4}$ · 3 years, 10 months ago