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1 Worksheet for Exploration 29.1: 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). 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. a. While you drag it, which way does the current in the loop flow? (right arrow means clockwise current; left arrow means counterclockwise current). Induced Current Direction= b. Sketch the field that the current in the loop generates. i. This may be called the induced field. c. In the center of the loop, does this field (created by the induced current) point into or out of the computer screen? Induced Field Direction at Center of loop= d. So, as you drag the loop to the right, the external field in the loop increases in which direction? The (induced )field generated by the (induced) current points in which direction? According to Lenz's law, these two directions should be opposite. i. Make a note to yourself that the change in the external field through the loop, was pointing in the same direction as the external field. ii. You should think about what is happening or what should happen as you drag the loop back to the original position (see if you can predict this before you do it below). Now, take the loop over to the far right and then move it slowly to the white region.

2 e. Explain why the direction of the current points the way it does. i. What is the direction of the initial external field pointing inside the loop? This can be drawn as a vector. ii. What is the change in the external field inside the loop as you move from blue into white (increase or decrease)? This is also a vector representing the change in external field inside the loop. iii. What is the direction of the induced field relative to the vector above in part ii? f. 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. i. You can always consider taking three steps as in part e. What is the direction of the external field on the inside of the loop initially? What is the direction of the change in the external field? What is the direction of the induced field? Once you know the direction of the induced magnetic field inside the loop, you can find the direction of induced current. g. 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? h. Try the two other configurations, Configurations A and B (where the magnetic field is hidden). Describe the magnetic field as completely as possible. i. For each configuration determine which way the field is pointing at various locations, and also if increasing or decreasing. Configuration A

3 Configuration B i. 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.

4 Worksheet for Exploration 29.2: Force on a Moving Wire in a Uniform Field a. What are the fluxes at t = 1 s and t = 3 s (from the graph)? Faraday's Law is a relationship between a time-varying magnetic field flux (Φ) and an induced emf (voltage), emf = - dφ/dt (position is given in meters, current is given in amperes, emf is given in volts, and magnetic flux is given in tesla meter 2 ). In this animation, a wire is pushed by an applied force in a constant magnetic field. Φ 1 = Φ 3 = b. What is the change in flux/second? ( Φ/ t). ( Φ/ t)= According to Faraday's law, this should be equal to the induced emf. c. Does your calculated emf agree with the emf reading on the meter connected to the wires? emf measured = d. What is the velocity of the sliding rod? v rod = e. What is the change in area/second? A/ t= f. Since Φ = B da, which is Φ = BA for this case (why?), what is the value of the magnetic field the wire slides in? i. Consider taking the derivative of both sides with respect to time.

5 The sliding wire has a current flowing in it. g. 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)? I graph (t= )= Direction: h. 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, F = IL x B. Direction of Force due to External Field= i. What is the value of the force? Magnitude of force on rod due to field= j. 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. i. Sketch a force diagram, and also indicate the net force acting 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. k. What is the power dissipated? Power=

6 The power delivered by an external force is W/ t, where W = F s is the work done by the applied force, F, and s is the displacement. l. Show that the power delivered is also F v. m. What is the power delivered by the external force? n. Why should this power be equal to the power dissipated by the circuit? o. Pick a different velocity and calculate the power dissipated by the circuit and the power delivered by the force.

7 Worksheet for Exploration 29.3: Loop Near a Wire A loop is near a wire which has a current flowing upwards. 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 s. a. How does the flux through the loop and the emf change as you drag the loop towards and away from the wire? Flux/toward/right side emf Flux/away/right side emf b. How does the flux through the loop and the emf change as you drag the loop parallel to the wire? Flux/parallel/right side

8 c. 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. Flux/toward/left side emf Flux/away/left side emf

9 Worksheet for Exploration 29.4: 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 T, and emf is given in millivolts). a. The vectors show the field through the loop as a function of time. What do the different colors indicate? b. What impact does changing the maximum value of the magnetic field have on the induced emf? i. For example what happens when you double the amplitude of the magnetic field? c. What impact does changing the frequency of the oscillation of the magnetic field have? i. Likewise again, consider doubling the frequency. d. Develop an expression to relate the change in the emf to the parameters you can vary. e. Develop an equation for the magnetic field as a function of time and the parameters you can vary. B= f. What is the area of the loop? Therefore, what is the flux through the loop as a function of time?

10 Flux(t)= g. Using Faraday's law, show that the emf should be equal to B max Aωcos(ωt + φ), where B max is the maximum value of the magnetic field in the x direction, A is the area of the loop, ω is the angular frequency of the oscillation, and φ is a phase angle. h. Verify that this expression matches the graph for the emf versus time.

11 Worksheet for Exploration 29.5: Self-inductance This animation shows a cross section of a solenoid (think of a long tube cut length wise 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. 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 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, H (1 H = 1 T m 2 /A). From Faraday s law, emf = - dφ/dt, the self-inductance is the back emf = - L (di/dt). This is a lot like for a capacitor, it took some effort or pushing of charge to charge it up. Here it takes some effort to establish a current. The inductance describes how difficult it is to establish the current (sounds like we do some work..we do). The back emf is in response to the magnetic field produced by the current being pushed through. So in our expression from Faradays law, the flux depends on magnetic field, which in turn depends on current and geometry. The geometry does not change. So the change in flux results from changing the current. All the rest of the stuff that determines the emf (geometry dependent stuff) is lumped together and called L (the self inductance). Recall that this is like capacitance, which also could be determined only from geometry, but could alternately be measured from the definition of capacitance. Faradays law for this case tells us how to measure L without messing around in the geometry. But just as for capacitors, if there is a simple geometry we can figure out how to predict what L should be for a given special case. 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). a. For the solenoid above, adjust the current with the slider and determine how the magnetic field varies with current. i. You should make several measurements of B vs. I. b. For this solenoid (given the value of the magnetic field at the current chosen), how many loops per meter are there? n=

12 Run change the current linearly as a function of time. c. What is the back emf? i. Note it is the back emf (or equivalently the induced emf) that is measured. The situation here is that an external emf source (battery) is applied to the solenoid directly. According to Kirchhoffs loop rule (conservation of energy) the sum of emf s and drops is zero. If resistances are small (battery and solenoid wire) then the back emf is equal and opposite to the applied or external emf. Even if the resistance is large, it can be measured and accounted for, thus allowing for a measure of the back emf. emf back = d. Using the equation for the back emf, what is the inductance, L? L measured = e. Using Faraday's law and the equation for the back emf, show that L = (Φ/ I) N for an inductor with N loops. f. Therefore, show that the inductance, L, of a solenoid is µ 0 N 2 A/(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/length is the number of loops per meter). g. If this solenoid is 2 m long, calculate the inductance and compare it to your answer in part (d) above. L predicted =

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