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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.
The wire shown above has its middle located at the origin. The wire has length \(L\) and a non-uniform charge density \(\lambda = \frac{q_o y}{L^2}\text{.}\)
For each of the two labeled points (A and B), is the electric potential positive, negative, or zero?
Tip.
Sketch a charge diagram of the wire.

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
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.
Tip.
Compare your two answers! You may find it useful to sketch a potential energy diagram to help interpret them.

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.
Write an expression for the electric potential at point A and point B for the wire above.

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 \(v\text{.}\)
What is the farthest distance along the \(y\)-axis that the small positive point charge will travel before it turns around?
Tip 1.
Draw a graph of the electric potential energy of this system vs. the \(y\)-position of the small positive point charge.
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!

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.)
Answer.
The first particle has 4 times the charge of the second particle.

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.
  1. What change in electric potential does the second particle experience?
  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.
Answer 1.
-5.0 V
Answer 2.
10.0 eV

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.
  1. What is the electric potential at the origin, which is midway between the charges?
  2. What is the electric potential at the position \(\vec{r} = (3.0 \mathrm{m})\hat{x} + (8.0 \mathrm{m})\hat{y}\text{?}\)
Answer 1.
14 V
Answer 2.
4.725 V

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?
Answer.
\(1.45 \times 10^{6} \mathrm{m}/\mathrm{s}\)

Calculation 23.4.9. Electron vs. Proton.

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

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?
Answer.
2560 V

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?
Answer.
156 \(\mu\)m

References References

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