5.4.3: Problems

Exercise \(\PageIndex{1}\): Change of electric potential in a uniform field

1. For each of the animations, does the potential energy of the particle increase or decrease as it moves?
2. Is the beginning point or ending point at a higher electric potential (higher voltage)?
3. Is an external force required or is the movement the result of the force due to the electric field? If an external force is required, describe it. Restart.

Problem authored by Morten Brydensholt modified by Anne J. Cox.

Exercise \(\PageIndex{2}\): Find the work done

An equipotential plot is shown in the animation (position is given in meters and electric potential is given in volts). The electric potential is shown next to the electron. You can drag the red electron with the mouse. How much work must an external force do in order to move the red electron from \([x,\: y] = [1.1\text{ m},\: -1.5\text{ m}]\) to \([x,\: y] = [-1.6\text{ m},\: 1.2\text{ m}\)]? Restart.

Problem authored by Mario Belloni and Wolfgang Christian and modified by Melissa Dancy.

Exercise \(\PageIndex{3}\): Rank the work done

An electron is moved through a region with an electric potential defined by the equipotential lines shown (position is given in meters, time is given in seconds, and electric potential is given in volts). Restart. Rank the work done by the external force (smallest to greatest) for Animations 1--5. Explain the basis for your ranking.

Problem authored by Melissa Dancy.

Exercise \(\PageIndex{4}\): Determine the unknown charge

The charges shown are given in units of 10^-10 C (position is given in meters and electric potential is given in volts). Restart. You can measure the electric potential by dragging the test charge around.

Determine the charge of the unknown charge for the following distributions:

1. Two charges.
2. Three charges.
3. Four charges.

Problem authored by Mario Belloni and Wolfgang Christian and modified by Anne J. Cox.

Exercise \(\PageIndex{5}\): Find mass of particle

In the animation the electric potential changes from \(0\text{ V}\) to \(1\text{ V}\) as shown by the equipotential lines (position is given in meters, time is given in seconds, velocity is given in meters/second, and electric potential is given in volts). Click-drag to place the \(1-\mu\text{C}\) test charge anywhere in the animation before you press "play." What is the mass of the particle? Restart.

Exercise \(\PageIndex{6}\): Draw the electric potential vs. \(x\) graph

Each animation shows a negative test charge moving under the influence of an electric potential (position is given in meters, time is given in seconds, and velocity is given in meters/second). Restart.

For each animation,

1. Sketch a graph representing the electric potential versus \(x\) for the region shown in the animation.
2. Sketch a graph representing the electric field versus \(x\) for the region shown in the animation.

Problem authored by Melissa Dancy.

Exercise \(\PageIndex{7}\): Determine the motion in a region of electric potential

The animations show a positive test charge in a region where there is an electric potential, as shown by the equipotential lines (position is given in meters, time is given in seconds, and electric potential is given in volts). The positive test charge is placed at \(x = 0\text{ m}\) with no initial velocity. Which animation most closely represents the charge's subsequent motion? Restart.

Problem authored by Melissa Dancy.

Exercise \(\PageIndex{8}\): Ranking fields and voltages

The animations show four charged conductors (position is given in meters and electric potential is given in volts). Restart.

1. Configuration 1: Rank the magnitude of the electric fields in the three regions between the conductors from largest to smallest. You can move the black circle around to show the voltage at any point.
2. Configuration 2: Rank the magnitude of the electric fields in the three regions between these conductors from largest to smallest.
3. Configuration 3: Rank the voltages on the four plates from largest to smallest. As you move the positive test charge around, the vector shows the strength of the force on the test charge.

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{9}\): Develop equation for voltage

Three point charges of equal charge are arranged on the corners of an equilateral triangle, as shown in the animation. Each charge is measured in \(\text{nC}\) (nanocoulombs or \(10^{-9}\text{ C}\)) and can be varied using the slider. You can click-drag to read the value of the voltage at any point in the animation (position is given in meters, voltage is given in volts, and charge is given in nanocoulombs). Restart.

1. Find a formula for the voltage midway between the charges (indicated by the small black dot) as a function of the charges.
2. Looking at the electric field vectors, where is the electric field zero? Explain.
3. Is the voltage at that spot also zero? If not, explain why you can have a zero electric field, but a nonzero voltage.

Problem authored by Chuck Bennett and Wolfgang Christian and modified by Anne J. Cox.

Exercise \(\PageIndex{10}\): Find unknown charge on pendulum

A charged ball with a mass of \(30\) grams is suspended from a string as shown. A second ball with a negative charge of \(3\text{ mC}\) is placed on a wood table. Play the animation and move the ball on the table toward and away from the pendulum (position is given in meters and time is given in seconds). Restart.

1. Find the equilibrium point of the pendulum for two different positions of the ball on the table.
2. Write an expression for the change in potential energy (both gravitational and electrostatic) from one equilibrium position to the other.
3. What is the charge on the pendulum? Assume that the charge on the two balls is uniformly distributed.

Exercise \(\PageIndex{11}\): Charge on a sphere

What is the total charge on the conducting sphere in this animation? Move the test charge to map out the magnitude of the electric potential as a function of distance from the center of the conducting sphere (position is given in centimeters, time is given in seconds, and electric potential is given in volts). The electric potential at infinity is \(0\) volts. Restart.

Problem authored by Anne J. Cox.
Script authored by Mario Belloni and Anne J. Cox.

Exercise \(\PageIndex{12}\): Cylindrical or spherical symmetry?

The animation shows either concentric cylinders or concentric spheres (position is given in centimeters and electric potential is given in volts). Restart. You can move the test charge around to measure the electric potential. Develop an equation for the voltage as a function of position for concentric cylinders and concentric spheres. The electric field outside a charged sphere \(= kQ/r^{2}\). Outside a charged rod it is \(2k\lambda /r\), where \(k = 9\times 10^{9}\text{ Nm}^{2}\) and \(\lambda =\text{ charge/length}\).

1. If this animation represented concentric spheres where the inner sphere has a voltage of \(10\text{ V}\) and the outer sphere has a voltage of \(0\text{ V}\), how much charge would be on the center sphere? What, then, would be the equation for the voltage as a function of position between the spheres?
2. If this animation represented concentric cylinders where the inner cylinder has a voltage of 10 V and the outer cylinder has a voltage of \(0\text{ V}\), what would the charge per unit length be on the inner cylinder? What, then, would be the equation for the voltage as a function of position between the cylinders?
3. Does this animation represent concentric cylinders or concentric spheres?

Problem authored by Anne J. Cox.
Script authored by Mario Belloni, Wolfgang Christian and Anne J. Cox.

 

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.