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Pre-Lab Quiz / PHYS 224

Electric Field and Electric Potential (B)

Your Name _________________________Lab Section _____________

1. What do you investigate in this lab?

2. For two concentric conducting rings, the inner radius of the larger

ring is 12.0 cm and the outer radius of the smaller ring is 4.0 cm. The

electric potential of the inner ring is 0 V and that of the outer ring

is 10 V. Calculate the electric potential half way between the two

rings, i.e., 8.0 cm away from the center of the rings.

3. In the following figure, the electric potential of the inner

conducting ring is 0 and the electric potential of the outer conducting

ring is −5.0 V. The two rings are concentric. Draw four equipotential

lines with 1 V-difference between two neighboring lines and label each

line with the corresponding voltage. Also draw four electric field lines

in each figure.

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Instructor’s Lab Manual / PHYS 224

Electric Field and Electric Potential (B)

Name _____________________________ Lab Section _____________

Objectives In this lab, you study the electric field between two coaxial metal

rings, map the equipotentials, determine the electric field, and plot

the electric field lines. The configuration resembles closely to a

uniaxial electric field.

Background

In electrostatics, the electric field (𝑬𝑬��⃗ ) and the electric

potential (𝑉𝑉) are two interchangeable physical quantities for describing

electric phenomena, although the former is a vector and the latter is a

scalar.

When placing a test electric charge q at a point P in space, it

experiences an electric force that is given by

𝑭𝑭��⃗ = 𝑞𝑞 𝑬𝑬��⃗ , (1)

where 𝑬𝑬��⃗ is the electric field at point 𝑃𝑃. When the test charge moves

from point 𝑃𝑃 to point 𝑄𝑄, the electric force on the charge is always

proportional to the electric field along the pathway. The total work

done by the electric force can be obtained by integrating the work along

the pathway. Because the electrostatic force is conservative, the total

work depends only on the starting and ending points of the pathway,

regardless on the specific pathway the charge traverses between those

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points. One can thus introduce the electric potential difference between

the two points, which is related to the total work (∆𝑊𝑊) done by the

electric force, as described by equation

∆𝑊𝑊 = −𝑞𝑞 ∆𝑉𝑉 = −𝑞𝑞 �𝑉𝑉𝑄𝑄 − 𝑉𝑉𝑃𝑃� (2)

Therefore, if one maps out the electric field in a certain region

of space, one can calculate the electric potential in that region (apart

from a common constant), and vice versa.

As a vector quantity, the electric field at a point must be

described by its magnitude and direction, altogether requiring three

numbers. To visualize 𝑬𝑬��⃗ in space, one normally draws electric field

lines (each line with an arrow to designate its direction) in space.

The direction of 𝑬𝑬��⃗ at a point is parallel to the tangent of the electric

field line passing through that point.

As a scalar quantity, the electric potential at any point requires

only one number to describe. As such, to plot the electric potential

in space, one can simply find and draw equipotential surfaces (often

referred simply as equipotentials). For every point of an equipotential

surface, the electric potential is the same.

Clearly, it is much easier to map and plot electric potential in

space. Normally, one draws a series of equipotentials with the same

potential difference between any two neighboring equipotentials. To

obtain the electric field from the plot of equipotentials, one uses the

following rules. At any point on an equipotential surface, the electric

field is always perpendicular to the equipotential surface, and the

electric field points from a high potential to a low potential. The

magnitude of the electric field is inversely proportional to the distance

between two neighboring equipotentials. If the potential difference

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between two infinitely close points 𝐴𝐴 and 𝐵𝐵 is ∆𝑉𝑉 = 𝑉𝑉𝐵𝐵 − 𝑉𝑉𝐴𝐴 and their

distance is ∆𝑑𝑑, the component of the electric field projected in the

direction pointing from A to B is given by �𝑬𝑬��⃗ � = ∆𝑉𝑉∆𝑑𝑑� .

A uniaxial electric field can be induced between two concentric

conducting cylindrical shells 𝑎𝑎 and 𝑏𝑏 with their longitudinal lengths

much larger compared to their separation. Assume the inner radius of

the larger ring as 𝑟𝑟𝑎𝑎 and the outer radius of the smaller ring as 𝑟𝑟𝑏𝑏.

When an electric potential difference ∆𝑉𝑉𝑜𝑜 = 𝑉𝑉𝑎𝑎 − 𝑉𝑉𝑏𝑏 is established

between the two rings, at any point between the rings the electric field

points radially and its magnitude depends only on the distance between

the point and the symmetry axis of the rings. By using calculus between

the rings and at a distance of 𝑟𝑟 from the symmetry axis of the rings,

the magnitude of the electric field is given by

𝐸𝐸 = 𝐶𝐶𝑟𝑟 (3)

where the constant 𝐶𝐶 = (𝑉𝑉𝑎𝑎 − 𝑉𝑉𝑏𝑏)𝑙𝑙𝑙𝑙 �𝑟𝑟𝑎𝑎

𝑟𝑟𝑏𝑏�� . Thus, equipotentials are also

concentric rings. The potential difference between two equipotentials is

however not linearly proportional to their distance. Between the rings

and at a distance of r from the symmetry axis of the rings, the electric

potential is given by

𝑉𝑉(𝑟𝑟) = 𝑉𝑉𝑏𝑏 + 𝐶𝐶 𝑙𝑙𝑙𝑙 � 𝑟𝑟𝑟𝑟𝑏𝑏�. (4)

The S.I. units for the above equations are: E in volts-per-meter

(V/m), d or r in meter (m), and V in volt (V).

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EXPERIMENT

Apparatus

Fill up with water the rectangular tray to a height of 0.5-inch.

Place the circular graph paper underneath the tray. Two metal rings are

used as electrodes to study uniaxially symmetric electric field. The

outer diameter of the smaller ring is 1.9 in and the inner diameter of

the larger ring is 6.0 in. Place the two metal rings inside the tray

and align their centers with the center on the graph paper.

Because the longitudinal lengths of the two metal rings are short

compared to their separation, the electric field between the two metal

bars cannot be considered exactly as uniaxial. Therefore, your

measurements in this lab may not completely follow Equations (3) and

(4).

Procedures Figure 1 displays the circular graph paper (not to scale) to be

used in this experiment. We will use polar coordinates (𝑟𝑟, θ) to identify

specific points. Note: the unit for the 𝑟𝑟 coordinate is inch.

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1. Set up the circuit (Figures 2-4)

Keep the power switch of the DC power supply in the off position.

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Follow Figure 2 to start the setup of the electrical circuit. i)

Place a point probe with a round tip (P1) over the inner metal ring.

Connect this point probe (P1) to the negative (ground) port of the DC

power supply. ii) Place a point probe with a round tip (P2) over the

outer metal ring. Connect this point probe (P2) to the positive terminal

of the DC power supply.

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As shown in Figure 3, iii) connect the point probe P3 to the common

port of Voltmeter A. Connect the handheld probe P4 to the voltage port

of Voltmeter A. Note: the sensitivity level of the voltmeter should be

set to 20 V (when you are asked to use it).

As shown in Figure 4, iv) connect probe P3 to the voltage input

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port of Voltmeter B. Connect the common port of Voltmeter B to probe

P1. Note: the sensitivity level of the voltmeter should be set to 20 V

(when you are asked to use it)

The point probes P1 and P2 will be used to set the potential across

the two metal rings, which will be provided by the DC power supply. The

point probe (P3) and the hand-held point probe (P4) will be used to map

potential through voltmeter A in the region between the rings. The tips

P1 and P2 should be always under water and in contact with the respective

metal ring. Make sure the two rings do not move during the experiment.

Also P3 and P4 probes should be under water when collecting data.

Ask your TA to check the circuit!

2. Investigate Electric Potential-versus-Distance

Turn on the DC power supply. Set the output voltage of the DC power

supply to 12 V.

Make sure Voltmeter B is off. Set the voltage level of Voltmeter A

to 20 V and turn it on. Place the tip of probe P3 at point (𝑟𝑟 = 1.5,

θ = 0°). Place the tip of probe P4 at points with θ = 0° and in turn with

𝑟𝑟′ = 1.7, 1.9, 2.1, 2.3, and 2.5. Read the corresponding potential difference

(∆ 𝑉𝑉) between the P4 and P3 probes from Voltmeter A. Record them in Table

1. Calculate ∆ 𝑉𝑉 /(𝑟𝑟 − 𝑟𝑟′) and record the ratio values in Table 1. (Note:

1 inch = 0.0254 m.)

TABLE 1

𝑟𝑟′

(inch)

Potential difference ∆𝑉𝑉

(V)

∆ 𝑉𝑉 /(𝑟𝑟 − 𝑟𝑟′)

(V/m)

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1.7

1.9

2.1

2.3

2.5

3. Map the first equipotential

Turn off Voltmeter A. Set voltage level of Voltmeter B to 20 V and

turn it on. Make sure the tip of probe P3 is at the point (𝑟𝑟 = 1.5, θ = 0°).

Read the potential at this point from Voltmeter B. Mark the point in

Figure 1. Record it in Table 2.

Turn off Voltmeter B and turn on Voltmeter A. Move the tip probe

P4 along the θ = 90° line until Voltmeter A reads 0 V. This means that

the tips of P3 and P4 are placed on the same equipotential. Read out the

𝑟𝑟 coordinate of the tip of P4 and mark the point in Figure 1. Record it

in Table 2.

Repeat the measurement by moving the tip of P4 respectively along

the θ = 180° and 270° lines until Voltmeter A reads 0 V. Read out the

corresponding 𝑟𝑟 coordinates of the tip of P4 and mark the corresponding

points in Figure 1. Record them in Table 2. Draw a curve through the

four points.

TABLE 2

Tip of P3

Measured

potential

(V)

𝑟𝑟 for

θ = 90°

𝑟𝑟 for

θ = 180°

𝑟𝑟 for

θ = 270°

Average

𝑟𝑟

Calculated

potential

(V)

(𝑟𝑟 = 1.5, θ = 0°)

(𝑟𝑟 = 1.7, θ = 0°)

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(𝑟𝑟 = 1.9, θ = 0°)

(𝑟𝑟 = 2.1, θ = 0°)

(𝑟𝑟 = 2.3, θ = 0°)

(𝑟𝑟 = 2.5, θ = 0°)

4. Map five more equipotentials

Repeat step 3 to map four more equipotentials by moving the tip of

probe P3 respectively to points (𝑟𝑟 = 1.7 , θ = 0°), (𝑟𝑟 = 1.9 , θ = 0°), (𝑟𝑟 =

2.1 , θ = 0°), (𝑟𝑟 = 2.3 , θ = 0°), and (𝑟𝑟 = 2.5 , θ = 0°).

Questions

1. For the measurement by step 2, should the measured ∆ 𝑉𝑉 be linearly

proportional to ∆ 𝑟𝑟?

2. Use an arrow to mark the directions of the electric field at (𝑟𝑟 =

2.0, θ = 45°) and (𝑟𝑟 = 2, θ = 225°) in Figure 1.

3. When using Equation (4) to describe the measured potential-versus-

radius, what are the values of 𝑟𝑟𝑎𝑎, 𝑟𝑟𝑏𝑏, 𝑉𝑉𝑎𝑎, 𝑉𝑉𝑏𝑏 , 𝑎𝑎𝑙𝑙𝑑𝑑 𝐶𝐶?

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Analysis Calculate the averaged 𝑟𝑟 coordinate for the four points on each

equipotential. Use the averaged 𝑟𝑟 coordinates and Equation (4) to

calculate the corresponding potential. Record them in Table 2 and compare

with the measured corresponding potentials.

Use the averaged 𝑟𝑟 coordinates and the measured corresponding

potentials to calculate the electric field values �𝑬𝑬��⃗ � = ∆𝑉𝑉∆𝑑𝑑� between

every two neighboring equipotentials. (Note: 1 inch= 0.0254 m.)

Between the 1st and 2nd: �𝑬𝑬��⃗ � =

Between the 2nd and 3rd: �𝑬𝑬��⃗ � =

Between the 3rd and 4th: �𝑬𝑬��⃗ � =

Between the 4th and 5th: �𝑬𝑬��⃗ �=

Between the 5th and 6th: �𝑬𝑬��⃗ �

In Figure 1, draw four electric field lines.