General Physics Lab 2 Siena College

Electric Fields and Potential

Object

This experiment further explores the electrostatic interaction between charged objects. The

concepts of electric field and potential are illustrated primarily through exercises with the EM

Field visualization program. Various stations provide additional hands-on demonstrations of

electric charge and force.

Equipment

This manual

EMField 6.4 Program

Theory

Fundamental forces interact via fields. In the case of electrostatics, fields arise from the potential

for a source charge to exert a force on a test charge, by virtue of the quantity of source charge

and the distance from it. We can re-write Coulomb’s Law for point charges in terms of the force

Fc between a test charge q0 and an electric field E;

FC q0 E (Eq. 1)

where both F and E are vectors. From Coulomb’s Law and Eq. 1, the expression for the electric

field of a point charge is given by

E kq

r2 r . (Eq. 2)

As this is a vector quantity, the field associated with a distribution of charges is determined from

the trigonometric vector sum of the fields associated with each source charge.

The force and field equations are analogous to those describing gravitational interactions.

Continuing the analogy, the work W associated with moving a charge q0 along an electric field

line is the scalar product of the Coulomb force with the distance:

W Fd q0Ed . (Eq. 3)

Recall that work is also defined as the change in energy of the test particle. In the case of

electrostatics, the particle is not moving and thus the energy is only potential energy, similar to

changing the gravitational potential energy by raising or lowering an object. The change in

potential energy is thus:

PE W q0Ed . (Eq.4)

2

We define a scalar quantity to describe the potential for interaction associated only with the

source charges, independent of the properties of any test charge. The difference in electric

potential, ΔV, is thus defined by dividing the difference in potential energy by the test particle

charge,

VPE

q0

. (Eq. 5)

This difference is more commonly known as the voltage between two points in an electric field.

The potential and potential energy are closely related but easily confused. The major difference

is that potential is associated with just the source independent of the test charge, whereas

potential energy involves both source and test charges.

NOTE: Eq. 3 and the right-hand equality in Eq. 4 are applicable only when E is constant or the distance d is very

small. Eq. 5 and the left-hand equality in Eq. 4 are correct in all cases.

At any distance r from a point charge, the potential is given by

V kq

r (Eq. 6)

assuming that V approaches zero as r approaches infinity. The potential energy between two

charges is thus

PE q2V1 kq1q2

r (Eq. 7)

Finally, points surrounding a charge at the same potential constitute an equipotential surface. In

the case of a point charge, the equipotential surfaces would simply be spheres enclosing the

charge at various distances. Such surfaces can be drawn like a topographical map to indicate the

strength of the electric field surrounding the source charge.

Part I: EM Field Program

In this exercise, you will explore the fields and potentials associated with various source charges.

Keep in mind that the field lines represent the force that a positive test charge experiences.

Electric field - definition in terms of force on a test charge

The electric field due to one or more charges is defined in terms of the force produced on a

positive separate charge.

In the diagram below, a point charge of + 4 units is placed at a grid point as shown. The "X"

indicates the location of a positive test charge with charge = +1 unit.

3

o o o o o o o o o o o o o

o o o o o o o o o o o o o

o o o o o X o +4 o o o o o

o o o o o o o o o o o o o

o o o o o o o o o o o o o

1. On the diagram, draw a vector to indicate the force on the test charge when located at point x.

Explain briefly how you arrived at the answer.

2. How would your answer change if the test charge was +2 units (instead of 1 unit)? Explain.

(Hint: consider Coulomb's law).

3. How would your answer change if the test charge was +3 units (instead of 1 unit)? Explain.

4. For each case considered, determine the ratio of the magnitude of the force felt by the test

charge to the size of the test charge. This ratio is (by definition) the magnitude of the electric

field due to the point charge.

5. Does the magnitude of the electric field depend on the size of the test charge used to measure it?

NOTE: The direction of the electric field is (by definition) the same as the direction of the force

on a positive test charge.

4

Electric field due to a single point charge

1. A point charge of +4 units is placed at a grid point in the diagram below. In the diagram, draw

a vector at each point marked "X" to indicate the electric field E at that point due to the point

charge. NOTE: Your vectors must properly display both the relative magnitudes and directions

of the electric field.

o o o o o o o o o o PREDICTION

o o o o o o o o o o o o o

o o o o o o o X o o o o o

o o o o o o o o o o o o o

o o o o o o X +4 o o X o o

o o o o o o o o o o o o o

o o o o o o o o o o o o o

o o o o o o o o o o o o o

2. How would your E vectors change if the point charge had charge of -4 units (rather than +4

units)? Explain briefly.

3. How would your E vectors change if the original charge (of +4 units) was moved one grid

point to the right?

5

Electric field due to a single point charge using EM Field

Procedure: Start the program EM FIELD 6.9. When the information screen appears, click

anywhere to run the program. To control this program, you will use the mouse to choose items

from the menu bar at the top of the white screen.

In the Sources menu, choose 3D point charges. Choose Show Grid from the Display menu,

and then Constrain to Grid from that same menu.

Drag a single point charge of +4 units (the solid circles) to the center of the grid.

1. While holding the mouse button down and not releasing it, slide the mouse across the screen

and around the point charge. Describe what you see on the screen.

(NOTE: Do not click with the mouse at this point. When you are finished, drag downward with

the mouse so that the cursor (actually an arrow) goes off the screen at the bottom.)

2. What does the vector indicate? Explain both magnitude and direction.

3. Refer to the previous grid diagram where you predicted the electric field at various points

(Step 1 of the previous section). At each location indicated by X, click with the mouse button.

Describe what you see on the screen. Is it what you predicted? If there are differences between

what you predicted and what you see on the screen, explain and resolve those differences.

4. Replace the +4 point charge with a -4 point charge at the same location. Refer to the previous

grid diagram where you predicted the electric field at various points (Step 2 of the previous

section). At each location indicated by X, click with the mouse button. Describe what you see on

the screen. Is it what you predicted? If there are differences between what you predicted and

what you see on the screen, explain and resolve those differences.

5. Replace the -4 point charge with a +4 point charge at the same location. Refer to the previous

grid diagram. At each location indicated by X, click with the mouse button. This should

duplicate the electric field vectors observed in step 2 above. Now move the +4 point charge one

grid point to the right by clicking, dragging, and releasing it at the desired spot. Compare what

you see on the screen with your prediction in question 3 of the previous section. If there are

differences between what you predicted and what you see on the screen, explain and resolve

those differences.

6

6. Choose Clean up screen from the Display menu in EM Field. This will remove electric field

vectors previously displayed. Identify three points that are 1, 2 and 3 grid units from the +4 point

charge. Evaluate the expression for the magnitude E of the electric field of a point charge: E =

kq/r2 to predict the magnitudes of E at the three points selected. Assume that k = 1 and that r is

expressed in units equal to the spacing between adjacent grid points.

7. Now click at the three points on the screen where you just predicted the magnitude of E. For

each point, measure the length of E in cm. How does this compare to the value of E you

predicted in the previous step?

8. Choose Field lines from the Field and potential menu. Click on the three points where the

electric field is displayed. EM Field will draw the electric field line through each of those points.

What is the relation between these electric field lines and the electric field vectors displayed in

the previous step? Click on a few more points surrounding the point charge, so that you have a

symmetrical pattern of electric field lines. Print the figure you have created on the screen by

choosing "Print screen" from the File menu.

7

Electric Potential: Work and Potential Difference

o o o o o o o o o o XD o o

o o o o o o o o o o o o o

o o o o o o o o o o o o o

o XA o o o o o XB o o +9 o o

o o o o o o o o o o o o o

o o o o o o o o o o o o o

o o o o o o o o o o XC o o

A. Suppose a charge q0 is located at position xA. The charge is able to move along any path to

positions xB, xc, and xD.

Make a prediction about the work done on the charge q0 to move it from rest at xA to rest at

xB, xc, or xD. Explain how you arrived at your answer. If there is not enough information for

you to answer the question, explain what information you would need.

B. Suppose the charge q0 is moved slowly by an external agent (e.g. a hand) along a straight

path from rest at xA to rest at xB.

1. Consider the force exerted by the external agent to move the charge. Is the work done

by this force positive, negative, or zero? Explain.

Compare the sign and magnitude of the work done by the electric force to the sign

and magnitude of the work done by the external force. Explain.

2. Write an equation that describes the work done by the electric field of the point

charge on the charge q0 as it moves from rest at xA to rest at xB. Explain.

8

3. Write an equation that describes the work done by the external agent on the charge q0

as it moves from rest at xA to rest at xB. Explain how you arrived at your answer.

4. How, if at all, would your answer to question 3 change if the charge had a magnitude

of 2q0?

How, if at all, would your answer to question 3 change if the charge had a magnitude

of 3q0?

Compare the ratios of the work done by the external force on the charge to the

magnitude of the charge.

The ratio you have found is the electric potential difference, ΔV. The electric potential difference

between two points describes the ratio of the work done on a test charge to move it from rest at

one point to rest at the second point to the magnitude of the test charge.

5. Does the potential difference depend on the magnitude of the test charge? Explain.

Consider that the charge is moved from rest at xA to rest at xB along the shortest possible path.

6. Compare the sign and magnitude of the potential difference for this path to the sign

and magnitude of the potential difference when the charge is moved from xB to xA.

Explain.

7. Find a value (in terms of k) for ΔV between locations xA (9 grid points from the q =

+9 charge) and xB (3 grid points from the q = +9 charge). Assume the units of charge

and distance are Coulomb and meter, respectively. Show all work.

9

Potential Difference of a Point Charge

A. Choose Clean up screen from the Display menu in EM Field. Select

Sources/3DPointCharges from the program’s menu. Place a single positive charge of

magnitude +9 at a grid point near the right side of the screen (about 2/3 of the way over) and

about half way down. Select Potential Difference from the Field and Potential menu.

1. WHILE HOLDING THE MOUSE BUTTON AND NOT RELEASING IT, slide the

mouse from the left edge of the screen toward the charge of +9 units. RELEASE the

mouse at a point close to but not on top of the charge. Describe what you see.

Choose Display/CleanUpScreen. Click and hold the mouse button and from xA to xB along a

straight path, releasing the mouse at xB.

o o o o o o o o o o XD o o

o o o o o o o o o o o o o

o o o o o o o o o o o o o

o XA o o o o o XB o o +9 o o

o o o o o o o o o o o o o

o o o o o o o o o o o o o

o o o o o o o o o o XC o o

2. Compare the value for the potential difference displayed on the computer with the

value you computed in question 7 on the previous page. Resolve any discrepancies.

3. Clear the screen again. Now click and hold and drag the mouse from xB to xA,

releasing the mouse button at xA. How does the value for ΔV between xB and xA

compare to the value between xA and xB? Explain.

How is the sign of ΔV reflected in the color of the line drawn on the screen?

4. Does the value of the potential difference between xB and xA depend on the path

along which you move? Explain how you arrived at your answer.

10

B. Let’s now consider positions xC and xD as well.

1. Is the potential difference between positions xA and xC greater than, less than, or

equal to the potential difference between positions xA and xB? Explain.

Choose Display/CleanUpScreen. Use the mouse to find the potential difference

between positions xA and xC. Compare the value of ΔV between xA and xC to the

value of ΔV between xA and xB. Resolve any discrepancies with your answer above.

Does the value of ΔV between xA and xC depend on the path you take to move from

xA and xC?

2. How, if at all, could you move a particle with charge q0 so that the work done on the

particle is always zero? Explain.

How, if at all, could you draw a path between points xB and xC along which the

potential difference is always zero? Explain.

Choose Display/CleanUpScreen. Use the mouse to find a path along with ΔV is

always zero. Compare this to your prediction. Resolve any discrepancies.

3. Predict the potential difference between points xA and xD (in terms of k).

4. Use the program to check your prediction. Resolve any discrepancies.

11

Field and Potential of a Dipole

A. Place a +9 and -9 charge two grid units apart near the center of the screen.

1. Draw the two charges and roughly sketch a circle around them. Predict the direction of

the dipole field at the top, bottom and sides of the circle. Also predict the field between

the charges.

2. Use the program to check your answer. Choose FieldAndPotential/FieldLines. Click to

draw field lines. Compare these field lines to your prediction. Resolve any discrepancies.

3. Now exam the electric potential around the dipole. First clean up field lines by selecting

Display/CleanUpScreen. Choose FieldAndPotential/Equipotentials With Number.

Click to draw equipotential lines between and around the charges. Explain qualitatively

the shape and relative strength of equipotential lines. How are the equipotential lines

oriented relative to the field lines?

Field and Potential of a Charged Plane

A. To create a charged plane, choose Sources/2DChargedRods. Create a horizontal line of

charges across the middle of the screen by dragging charged rods of charge equal 1. Adjacent

rods should be touching. The length of the rods is into the screen (in the z direction if x and y

are the plane of the screen), so you now have a plane of charge.

1. Choose FieldAndPotential/FieldLines. Click to draw field lines above and below

the charged plane. Print a picture and explain the pattern of the field lines.

2. Now examine the electric potential around the charged plane. Choose

FieldAndPotential/Equipotentials With Number. Click to draw equipotential lines

above and below the charged plane. Explain qualitatively the shape and relative

strength of equipotential lines. How are the equipotential lines oriented relative to the

field lines?

12

A capacitor consists of two oppositely charged plates. To examine the field and potential

inside a capacitor, create another horizontal line of charge 3 units above your first line.

Again, choose Sources/2DChargedRods . Use -1 charge to create this second line.

3. Now examine the electric field between these two charged planes. Choose

FieldAndPotential/FieldLines. Click to draw field lines above, below, and between

the charged planes. Print a picture and explain the pattern of the field lines.

4. Now examine the electric potential between the charged planes. Choose

FieldAndPotential/Equipotentials With Number. Click to draw equipotential lines

above, below, and between the charged planes. Provide a qualitative explanation for

the shape and relative strength of the equipotential lines. How are the equipotential

lines oriented relative to the field lines?