Superposition of electric fields

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Name	Status	Author	Last Modified
Superposition of electric fields	Ready to use	Jon Goss	07/04/2021 12:35
Superposition of electric fields	Ready to use	Jon Goss	01/02/2021 14:42

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Question statement

A square of side length {sidecm} cm has charges of different magnitudes placed at its four corners. Clockwise from the top right as viewed from above these are $q_1$ to $q_4$ , with values {q1}, {q2}, {q3} and {q4} nC. In this question, you are required to determine the total electric field, including both magnitude and direction, at the centre of the square due to this arrangement of point charges.

This question is a relatively complicated example of application of the principle of superposition.

You should use the values of the physical constants provided in the module notes, assume the medium is air, quote all answers to 4 significant figures and you may express values in scientific format.

Give any introductory information the student needs.

			Name	Type	Generated Value

Emag	decimal	dec("2.207365688499963238101326272508069854741e+4")
eps0	number	0.000000000008854
epsair	number	1.0005
Ex1	decimal	dec("8.39949430304296848680195848395454545455e+3")
Ex2	decimal	dec("-9.449431090923339693748645688363636363634e+3")
Ex3	decimal	dec("-3.149810363641113620840061613227272727274e+3")
Ex4	decimal	dec("-5.249683939401855333470512531254545454544e+3")
Extotal	decimal	dec("-9.449431090923340161257261348890909090902e+3")
Ey1	decimal	dec("8.39949430304296848680195848395454545455e+3")
Ey2	decimal	dec("9.449431090923339693748645688363636363634e+3")
Ey3	decimal	dec("-3.149810363641113620840061613227272727274e+3")
Ey4	decimal	dec("5.249683939401855333470512531254545454544e+3")
Eytotal	decimal	dec("1.994879896972704989318105509034545454545e+4")
k	number	8983250812.581928
q1	integer	-8
q2	integer	9
q3	integer	-3
q4	integer	-5
qq1	number	-0.000000008
qq2	number	9.000000000000001*10^(-9)
qq3	number	-3.0000000000000004*10^(-9)
qq4	number	-0.000000005
r	decimal	dec("7.778174593052023036818575966306688126535e-2")
rsq	decimal	dec("6.050000000000000417546860865949403817418e-3")
side	rational	11/100
sidecm	integer	11
theta	decimal	dec("1.965382405805330255479846784865504038131e+1")
thetad	decimal	dec("1.965382405805330255479846784865504038131e+1")
thetar	decimal	dec("3.43023940420703270228957933126075797451e-1")

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
sqrt(Extotal^2+Eytotal^2)

JME function reference

Description

Magnitude of the field at the centre.

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `decimal`

dec("2.207365688499963238101326272508069854741e+4")

← Depends on:

Extotal
Eytotal

→ Used by:

thetar

Parts

Field due to first charge Gap-fill
1. Gap Ex1 Number entry
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2. Gap Ey1 Number entry
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Field due to second charge Gap-fill
1. Gap Ex2 Number entry
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2. Gap Ey2 Number entry
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Field due to third charge Gap-fill
1. Gap Ex3 Number entry
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2. Gap Ey3 Number entry
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Field due to fourth charge Gap-fill
1. Gap Ex4 Number entry
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2. Gap Ey4 Number entry
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Total field Gap-fill
1. Gap Extotal Number entry
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2. Gap Eytotal Number entry
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Angle Gap-fill
1. Gap theta Number entry
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Gap-fill

Ask the student a question, and give any hints about how they should answer this part.

Determine the electric field at the centre of the square due to $q_1$ ({q1}nC).

$E_x(q_1)=$ Ex1V/m

$E_y(q_1)=$ Ey1V/m

Advice

This is a problem of superposition of four different fields, which amounts to the sum of four vectors. Each field is an expression of Coulomb's Law, so that the pre-factor is

$\displaystyle {q_i\over 4\pi\varepsilon_0 r^2}$ ,

where since the point of interest is the centre of the square, the distance $r$ is the same for each point-charge (in this case {side*100}/ $\sqrt{2}\approx$ {siground(r*100,4)} cm). In each case the field points radially away from the point-charge so that for $q_1$ it is down and left in equal measures, for $q_2$ it is up and to the left, and so on. For negative charges the sign of $q$ can be thought of reversing the direction (fields converge on negative point charges). Take care in determining the direction and note that diagrams generally help!

The value of $E$ can be readily obtained and the direction in each case given by

$\displaystyle{1\over\sqrt{2}}\left(\pm \hat{i} \pm \hat{j}\right)$ ,

with the combinations of $\pm$ depending upon which charge we're considering and the $\sqrt{2}$ factor coming from the fact that this is a unit vector.

Once each vector field has been obtained they can be summed by component (apply the principle of superposition) giving the total $E$ -field at the centre of the square.

Finally the angle with the line joining $q_2$ and $q_4$ can be obained by noting that the dot product of the total field with the direction of the displacement from $q_2$ to $q_4$ yields an angle:

$\displaystyle{\vec{E}.\left(-\frac{\hat{i}}{\sqrt{2}}+\frac{\hat{j}}{\sqrt{2}}\right)=\left|\vec{E}\right|\cos(\theta)}$

$\displaystyle{-E_x+E_y=\sqrt{2}\left|\vec{E}\right|\cos(\theta)\Rightarrow \theta=\cos^{-1}\left(\frac{E_y-E_x}{\sqrt{2}\left|\vec{E}\right|}\right)}$ .

It then only remains to determine the value of theta that lies in the required angle range.

Give a worked solution to the whole question.

URL

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This question is used in the following exams:

Tutorial questions - electrostatics by Jon Goss in EEE1001 & PHY1023.
All electromag by Jon Goss in EEE1001 & PHY1023.
TEST by Jon Goss in Jon's workspace.
E&M digital tutorial week 3 by Jon Goss in EEE1001 & PHY1023.
NCM4 FE-F2022 UNIT 8 All by Terry Young in NCM4 FINAL EXAM Fall 2022.

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Jon Goss 5 years, 4 months ago

Jon Goss 5 years, 5 months ago

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Superposition of electric fields

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Jon Goss 4 years, 2 months ago

Jon Goss 5 years, 4 months ago

Jon Goss 5 years, 5 months ago

Jon Goss 5 years, 5 months ago

Error in variable testing condition

Error

Warning

Generated value: decimal

Parts

Gap-fill

Generated value: `decimal`