5.8.2: Explorations

Exploration 1: Lenz's Law

Lenz's law is the part of Faraday's law that tells you in which direction the current in a loop will flow. Current flows in such a way as to oppose the change in flux. The magnetic field created by the current in the loop opposes the change in the magnetic flux through the loop's area (position is given in meters, time is given in seconds, and magnetic field strength is given in tesla). Restart.

Consider the initial configuration. The center has a field-free region, and the sides have a linearly increasing magnetic field into the computer screen (blue) and out of the computer screen (red). The deeper colors represent a stronger field. Drag the loop from the white (field-free region) into the blue.

1. While you drag it, which way does the current in the loop flow? (right arrow means clockwise current; left arrow means counterclockwise current).
2. Sketch the field that the current in the loop generates.
3. In the center of the loop, does this field (created by the induced current) point into or out of the computer screen?
4. So, as you drag the loop to the right, the external field in the loop increases in which direction? The field generated by the current points in which direction? According to Lenz's law, these two directions should be opposite.

Now, take the loop over to the far right and then move it slowly to the white region.

5. Explain why the direction of the current points the way it does.
6. What if you take the loop from the center to the left (into the red region)? Explain what you expect to happen and then try it.
7. Can you tell the difference between moving a loop from a blue to a white region and moving from a white region to a red region? Why or why not?
8. Try the two other configurations, Configurations \(A\) and \(B\) (in which the magnetic field is hidden). Describe the magnetic field as completely as possible.
9. Once you've completed your descriptions, decide which of the magnetic fields (Fields \(1,\: 2,\) or \(3\)) matches Configuration \(A\) and Configuration \(B\).

Check your answers to (i) by adding a loop to a field animation:

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

Exploration 2: Force on a Moving Wire in a Uniform Field

Faraday's Law is a relationship between a time-varying magnetic field flux (\(\phi\)) and an induced emf (voltage), \(\text{emf }= -\Delta\phi /\Delta t\) (position is given in meters, current is given in amperes, emf is given in volts, and magnetic flux is given in tesla per meter\(^{2}\)). In this animation, a wire is pushed by an applied force in a constant magnetic field. Restart.

1. What are the fluxes at \(t = 1\text{ s}\) and \(t = 3\text{ s}\) (from the graph)?
2. What is the change in flux/second (\(\Delta\phi /\Delta t\))?

According to Faraday's law, this should be equal to the induced emf.

3. Does your calculated emf agree with the emf reading on the meter connected to the wires?
4. What is the velocity of the sliding rod?
5. What is the change in area/second?
6. Since \(\phi =\int\mathbf{B}\cdot d\mathbf{A}\), which is \(\phi = BA\) for this case (why?), what is the value of the magnetic field the wire slides in?

The sliding wire has a current flowing in it.

7. In what direction is this current and what is the value of the current (read the current value from the graph) at a given time (pick a time)?
8. In what direction is the magnetic force on this current-carrying wire moving in the external magnetic field [the one you found in part (f) above]? Remember, \(\mathbf{F}=I\mathbf{L}\times\mathbf{B}\).
9. What is the value of the force?
10. Since the wire moves at a constant speed, what must be the direction and magnitude of the applied force? Check your answer by showing the force on the wire.

The power dissipated in an electrical circuit is the current times the voltage drop. In this case, \(I\) times the emf across the rod.

11. What is the power dissipated?

The power delivered by an external force is \(\Delta W/\Delta t\), where \(W =\mathbf{F}\cdot\mathbf{s}\) is the work done by the applied force, \(\mathbf{F}\), and \(\mathbf{s}\) is the displacement.

12. Show that the power delivered is also \(\mathbf{F}\cdot\mathbf{v}\).
13. What is the power delivered by the external force?
14. Why should this power be equal to the power dissipated by the circuit?
15. Pick a different velocity and calculate the power dissipated by the circuit and the power delivered by the force.

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

Exploration 3: Loop Near a Wire

A loop is near a wire that has a current flowing upward. You can drag the loop (position is given in meters, magnetic field strength is given in millitesla, emf is given in millivolts, and time is given in seconds). The flux through the loop and the induced emf are shown in the graph. The animation will stop after \(30\text{ s}\). Restart.

1. How do the emf and the flux through the loop change as you drag the loop toward and away from the wire?
2. How do the emf and the flux through the loop change as you drag the loop parallel to the wire?
3. Are the flux and emf different when the loop is on the left side, instead of the right side, of the current-carrying wire? Explain.

Exploration authored by Melissa Dancy.

Exploration 4: Loop in a Time-Varying Magnetic Field

The animation shows a wire loop in a changing magnetic field. The graphs show the magnetic field in the \(x\) direction as a function of time and the induced emf in the loop (position is given in meters, magnetic field strength is given in millitesla, \(10^{-3}\text{ T}\), and emf is given in millivolts). Restart.

1. The vectors show the field through the loop as a function of time. What do the different colors indicate?
2. What impact does changing the maximum value of the magnetic field have on the induced emf?
3. What impact does changing the frequency of the oscillation of the magnetic field have?
4. Develop an expression to relate the change in the emf to the parameters you can vary.
5. Develop an equation for the magnetic field as a function of time and the parameters you can vary.
6. What is the area of the loop? Therefore, what is the flux through the loop as a function of time?
7. Using Faraday's law, show that the emf should be equal to \(|B_{\text{max}}|A\omega\cos(\omega t +\phi)\), where \(|B_{\text{max}}|\) is the maximum value of the magnetic field in the \(x\) direction, \(A\) is the area of the loop, \(\omega\) is the angular frequency of the oscillation, and \(\phi\) is a phase angle.
8. Verify that this expression matches the graph for the emf vs. time.

Exploration authored by Anne J. Cox.

Exploration 5: Self-Inductance

This animation shows a cross section of a solenoid (think of a long tube cut lengthwise down the cylinder and then looking at the edge) so that the black dots represent the current-carrying wires coming into and out of the screen. Restart. The arrows show the direction and magnitude of the magnetic field. You can drag the black dot around to measure the field in different spots (position is given in centimeters, the magnetic field strength is given in millitesla, \(10^{-3}\text{ T}\), and current is given in amperes). You can either change field by varying the current in the wires with the slider or you can choose to change the current linearly as a function of time.

Faraday's law tells us that when a loop is in a changing magnetic field, an induced emf in the loop will result. But what if the loop itself has a changing current? With a changing current, the loop has a changing magnetic field. Wouldn't it make sense, then, for there to be an induced emf and an induced current to oppose the changing flux? The answer is that there are: If the current is changed in a current loop, there is a self-induced back emf. The measure of the back emf produced when a current is changed in a loop is called its self-inductance, or simply inductance, represented by \(L\) and measured in henries, \(\text{H}\) (\(1\text{ H} = 1\text{ T}\cdot\text{m}^{2}\text{/A}\)). From Faraday's law, \(\text{emf }= - d\phi /dt\), the self-inductance is the back \(\text{emf} = - L (dI/dt)\).

Run the change field by varying the current in the wires with the slider. Instead of considering a loop, we will look at a solenoid (it is easier to calculate the magnetic field inside a long solenoid).

1. For the solenoid above, adjust the current with the slider and determine how the magnetic field varies with current.
2. For this solenoid (given the value of the magnetic field at the current chosen), how many loops per meter are there?

Run change the current linearly as a function of time.

3. What is the emf?
4. Using the equation above, what is the inductance, \(L\)?
5. Using Faraday's law and the equation above, show that \(L = (\phi /I) N\) for an inductor with \(N\) loops.
6. Therefore, show that the inductance, \(L\), of a solenoid is \(\mu_{0}N^{2}A/\text{(length)}\), where \(N\) is the number of loops, \(A\) is the cross-sectional area and length is the length of the solenoid (so that \(N/\text{length}\) is the number of loops per meter).
7. If this solenoid is \(2\text{ m}\) long, calculate the inductance and compare it to your answer in (d) above.

Exploration 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.