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Learning Introductory Physics with Activities

Paul J. Emigh, Rebecka Tumblin, Kathryn Hadley, Danielle Skinner

Section 23.4 Practice - Electric Energy and Potential

Subsection Explanation Practice

Explanation 23.4.1. Electric Potential of a Wire I.

Figure 23.4.1. A charged wire lies along the y-axis centered on the origin.
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The wire shown above has its middle located at the origin. The wire has length \(L\) and a non-uniform charge density \(\lambda = \frac{q_o y}{L^2}\text{.}\)
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For each of the two labeled points (A and B), is the electric potential positive, negative, or zero?
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Tip.
Sketch a charge diagram of the wire.
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Subsection A*R*C*S Practice

A*R*C*S 23.4.2. Moving Charges I.

Two point charges are fixed in place along the x-axis: \(-Q_o\) at \(x = +L\) and \(+4Qo\) at \(x = -L\text{.}\) A small positive point charge (\(+q_i\)) is initially located at \(x = 3L\text{,}\) which is the location where the net force is zero. The small positive charge is then given a tiny push either to the right or to the left
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For each case (tiny push to the right and tiny push to the left), determine the speed of the small positive charge when it has moved a distance \(L\) from its initial position.
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Tip.
Compare your two answers! You may find it useful to sketch a potential energy diagram to help interpret them.
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A*R*C*S 23.4.3. Electric Potential of a Wire II.

Figure 23.4.2. A charged wire lies along the y-axis centered on the origin.
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Write an expression for the electric potential at point A and point B for the wire above.
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A*R*C*S 23.4.4. Moving Charges II.

Two identical negative point charges (\(-Qo\)) are fixed in place at \(x = +L\) and \(x = -L\text{.}\) A small positive point charge (\(+q_i\)) with mass \(m\) is at the origin, moving in the negative y-direction with speed \(v\text{.}\)
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What is the farthest distance along the \(y\)-axis that the small positive point charge will travel before it turns around?
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Tip 1.
Draw a graph of the electric potential energy of this system vs. the \(y\)-position of the small positive point charge.
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Tip 2.
Evaluate your answer when the initial speed \(v\) increases. This one can be tricky—make sure you check that your answer matches what you expect to happen!
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Subsection Numerical Practice

Calculation 23.4.5. Electric Potential Comparison.

The electric potential at a location 1 m from a certain charged particle is 4 times the potential at a location 1 m from a different particle. How do the particles’ charges compare? (These are two separate trials; assume that only one particle is present for each trial.)
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Answer.
The first particle has 4 times the charge of the second particle.
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Calculation 23.4.6. Moving Electron.

An electron moves through a potential difference of -5.0 V. A second particle with twice the negative charge moves between the same two points.
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  1. What change in electric potential does the second particle experience?
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  2. What change in electric potential energy does the second particle experience? (Note: The units used here are eV which are electron-volts. 1 electron-volt is the energy gained/lost by an electron when it goes through a potential difference of 1 volt.
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Answer 1.
-5.0 V
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Answer 2.
10.0 eV
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Calculation 23.4.7. Electric Potential due to Two Point Charges.

Consider two point charges of equal value, \(+(7/3) \times 10^{-9}\) C, located along the x-axis. One charge is placed at x = -3.00 m, the other at x = 3.00 m.
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  1. What is the electric potential at the origin, which is midway between the charges?
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  2. What is the electric potential at the position \(\vec{r} = (3.0 \mathrm{m})\hat{x} + (8.0 \mathrm{m})\hat{y}\text{?}\)
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Answer 1.
14 V
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Answer 2.
4.725 V
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Calculation 23.4.8. Release the Electron.

An electron is initially at rest at a location where the electric potential is -4 V. When released from rest, the electron accelerates, passing through a location where the electric potential is 2 V. What is the speed of the electron when it passes through that second location?
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Answer.
\(1.45 \times 10^{6} \mathrm{m}/\mathrm{s}\)
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Calculation 23.4.9. Electron vs. Proton.

Consider a charge moving through a region of space where the electric potential varies.
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  1. Which of the following statements are true if the charge is a proton?
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    1. The proton is driven toward the region of lowest potential energy.
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    2. The proton is driven toward the region of highest potential energy.
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    3. The proton is driven toward the region of lowest electric potential.
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    4. The proton is driven toward the region of highest electric potential.
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    5. The proton experiences a force directed toward a lower potential.
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    6. The proton experiences a force directed toward a higher potential.
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  2. What of the following statements are true if the charge is an electron?
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    1. The electron is driven toward the region of lowest potential energy.
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    2. The electron is driven toward the region of highest potential energy.
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    3. The electron is driven toward the region of lowest electric potential.
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    4. The electron is driven toward the region of highest electric potential.
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    5. The electron experiences a force directed toward a lower potential.
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    6. The electron experiences a force directed toward a higher potential.
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Answer 1.
(a), (c), (e)
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Answer 2.
(a), (d), (f)
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Calculation 23.4.10. Potential Difference.

An electron is placed in an external electric potential field. Through what change in potential must this electron go in order to (starting from rest) reach a speed of 0.10 c, where c is the speed of light?
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Answer.
2560 V
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Calculation 23.4.11. Closest Approach.

Consider a proton at the origin of a coordinate system, fixed in place (not able to move at all). Now consider a second proton, free to move, that is launched directly at the first proton at a speed of 42 m/s from a distance very, very far away. What is the closest the second proton gets to the first proton?
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Answer.
156 \(\mu\)m
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References References

[1]
Practice activities provided by BoxSand: https://boxsand.physics.oregonstate.edu/welcome.
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