5.8.3: Problems

Exercise \(\PageIndex{1}\): Find direction of current

Three loops are shown in a region where the magnetic field is changing (position is given in meters and time is given in seconds). Blue indicates the magnetic field is directed into the screen and red indicates it is directed out of the screen. The intensity of the color represents the magnitude of the field. At any instant of time, the red and blue "fields" have the same magnitude. Restart.

For each of the following times, is there a current in each loop (\(A,\: B,\) and \(C\)) and, if there is a current, is it flowing clockwise or counterclockwise? Explain.

1. \(t = 0.5\text{ s}\).
2. \(t = 3.1\text{ s}\).
3. \(t = 4.0\text{ s}\).

Problem authored by Melissa Dancy.

Exercise \(\PageIndex{2}\): Graph current versus time

A loop is shown in a region where there is a magnetic field (position is given in meters and time is given in seconds). Blue indicates where the magnetic field is directed into the screen and red indicates where it is directed out of the screen. Restart.

1. Construct a qualitative graph of current induced in the loop vs. time. Indicate on your graph the time when the current reaches its maximum and minimum values. Consider the current to be positive when it flows clockwise and negative when it flows counterclockwise.
2. How would your graph be different if the region on the left side of the applet had zero magnetic field instead of a field into the screen? Explain.

Problem authored by Melissa Dancy.

Exercise \(\PageIndex{3}\): Find the force on a loop

A loop with a resistance of \(200\Omega \) is pulled at constant velocity through a region where there is a magnetic field of \(2\text{ T}\) out of the screen and into a region of no magnetic field (position is given in meters and time is given in seconds). Restart.

During the time shown in the animation,

1. What are the direction and magnitude of the current in the loop?
2. A force is necessary to pull the loop out of the magnetic field. Why? (Draw a free-body diagram of the loop and explain the origination of each force.)
3. Find the magnitude of the force that was exerted by the hand on the loop in the animation.
4. Describe the subsequent motion of the loop if the same force continues to act on the loop when it is in the region of no magnetic field.

Problem authored by Melissa Dancy.

Exercise \(\PageIndex{4}\): Rank currents in a loop

A loop, next to a very long current-carrying wire, moves along the path shown (position is given in meters and time is given in seconds). The trail of the loop is marked. The current in the wire is constant and is directed upward as shown by the arrow. Restart.

1. For \(t = 0.5\text{ s},\: 1.5\text{ s},\: 2.5\text{ s},\: 3.5\text{ s}\), and \(4.5\text{ s}\), is the induced current in the loop clockwise, counterclockwise, or zero?
2. Rank the magnitudes of the currents induced in the loop (smallest to greatest, explicitly indicating any ties) at \(t = 0.5\text{ s},\: 1.5\text{ s},\: 2.5\text{ s}\), and \(3.5\text{ s}\).

Problem authored by Melissa Dancy.

Exercise \(\PageIndex{5}\): Find emf in loop in changing magnetic field

The animation shows the cross section of a wire loop in a changing magnetic field (the wire loops out of and into the screen). The graph shows the magnetic field in the \(x\) direction as a function of time (position is given in centimeters, time is given in milliseconds, \(10^{-3}\text{ s}\), and magnetic field strength is given in tenths of tesla, \(10^{-1}\text{ T}\)). Restart.

1. Sketch a graph of the induced emf in the loop as a function of time.
2. What is the maximum value of the emf?
3. What is the direction of the current in the loop as a function of time (use the convention that current flowing out of the top of the loop is positive)?

Problem authored by Anne J. Cox.

Exercise \(\PageIndex{6}\): In which direction is current flowing?

A dragable conducting loop is to the right of a wire as shown in the animation (position is given in meters and time is given in seconds). The graph shows the voltage read by the voltmeter attached to the conducting loop. (The polarity is also shown: when the voltage is positive, it means the voltage is higher at the red terminal than at the blue one, so current flows from the red around the loop counterclockwise to the blue). In which direction is the current flowing in the long straight wire? Restart.

Problem authored by Mario Belloni and modified by Melissa Dancy and Anne J. Cox.

Exercise \(\PageIndex{7}\): Bar magnet inserted through a loop

The animation shows a top view of four wires and a galvanometer in the lab. Current flowing into the \(+\) terminal, i.e., counterclockwise, will deflect the meter to the right (positive voltage). During the time interval \(t = 2\text{ s}\) to \(t = 8\text{ s}\) a magnet is slowly pushed completely through the rectangle from below to above. The animation starts at \(t = 0\text{ s}\) and stops at \(t = 10\text{ s}\).  Observe the meter reading (shown on the graph) during this simulation. In the animation of moving magnets, you see a side view of the process. Think of the left side of the gray loop as behind the computer screen and the right side as in front of the screen. Restart.

Which of the animations of moving magnets best matches what happens to create the graph shown?

Problem authored by Wolfgang Christian and modified by Melissa Dancy and Anne J. Cox.

Exercise \(\PageIndex{8}\): Determine the direction of the magnetic field through the loop

The animation shows a top view of four wires and a galvanometer (position is given in meters and time is given in seconds). There is a constant magnetic field passing through the area enclosed by the wires. Current flowing into the \(+\) terminal, i.e., counterclockwise, will deflect the meter to the right. You can drag the black bar on the right. The animation runs from \(0\) to \(10\text{ s}\) and then repeats. Restart.

Determine the direction of the magnetic field through the loop.

Problem authored by Mario Belloni modified by Melissa Dancy.

Exercise \(\PageIndex{9}\): Find the magnetic field

The wire on the right is pulled to the right for \(5\text{ s}\) and then to the left for \(5\text{ s}\) as shown. Determine the magnitude of the magnetic field passing through the wire rectangle. You may pause the animation and read coordinate values by click-dragging the mouse at any location (position is given in centimeters and emf is given in microvolts, \(10^{-6}\text{ V}\)). Restart.

Problem authored by Mario Belloni and modified by Melissa Dancy.

Exercise \(\PageIndex{10}\): Find the field in a generator

The graph on the right shows the induced emf through the loop as a function of time (position is given in centimeters, time is given in seconds, and emf is given in millivolts). The green arrow shows the direction and magnitude of the induced current. Restart.

1. What is the magnitude of the magnetic field?
2. Looking down on this loop from above, in what direction is it rotating (clockwise or counterclockwise)? Explain your answer.

Problem authored by Melissa Dancy and Anne J. Cox.
Script authored by Wolfgang Christian, Anne J. Cox, and Melissa Dancy.

Exercise \(\PageIndex{11}\): Find the inductance of a solenoid

This animation shows a cross section of a solenoid (think of a pipe cut in half lengthwise and looking at the cut edge). The graphs show the current in the solenoid and the induced emf as a function of time (position is given in centimeters, time is given in seconds, magnetic field strength is given in millitesla,  current is given in amperes, and induced emf is given in millivolts). Restart.

1. What is the inductance of the solenoid shown in this animation?
2. How many turns per meter of current-carrying wire are coiled around this solenoid?
3. How long is this solenoid (it extends beyond the screen)?

Problem authored by Anne J. Cox.

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