Magnetism and Induction

Transcription

1 Magnetism and Induction Before the Lab Read the following sections of Giancoli to prepare for this lab: 27-2: Electric Currents Produce Magnetism 28-6: Biot-Savart Law EXAMPLE 28-10: Current Loop 29-1: Induced EMF 29-2: Faraday s Law of Induction; Lenz s Law This lab is an introduction to magnetism and inductance, and it may (depending on when you perform the lab) contain material that you have not yet encountered in class. Consequently, it is very important that you read the material that is listed above before you come to the lab. Introduction and Theory Electric currents give rise to magnetic fields. However static magnetic fields, and by this we mean fields that do not change in time, do not generate electric currents. It was Michael Faraday who first discovered that a changing magnetic field is required to induce an electric current to flow. This phenomenon is called electromagnetic induction. It is of enormous practical importance and used in transformers and generators. In this lab you will use electromagnetic induction to measure the direction and magnitude of the magnetic field produced by a loop of alternating current. This will give us an opportunity to study a system in the lab experimentally that we study in class theoretically. The magnitude of the voltage E induced in the pick-up coil by a changing magnetic field in

2 2 the field coil is described by Faraday s Law: E = N Φ B. (1) Where: N is the number of loops of wire in the pick-up coil and Φ B is the rate of change of the magnetic flux through the pick-up coil. The minus sign indicates that the direction of the induced current is opposite to the direction of the current that produced the magnetic field (Lenz s law). The magnetic flux Φ B is defined using the magnetic field B, the surface vector of the pick-up coil A (equal to the surface area times the surface normal vector A A ˆn), and the angle θ between B and A: Φ B = B A = BA cos θ. (2) In class, the concept of flux is first introduced when we discuss Gauss s Law. The flux of a vector field gives us a quantitative measure of both the strength of the field (the number of field lines) and also how the field lines are oriented in space relative to the surface A. If the field lines pass through (or cut) the surface perpendicularly the flux is maximum and equal to BA because cos θ = 1. If the field lines lie in the plane of the surface, and do not cut the surface, the flux is zero because cos θ = 0. To induce a current in the pick-up coil we pass an alternating or a.c. current through the large coil called the field coil. When you meet alternating currents for the first time, which you probably are now, they appear quite strange. We can easily find analogues for direct current (d.c. current) where the direction of current flow does not change. Water flowing through a pipe or cars traveling down a highway are both good analogues for d.c. current flow. But in an alternating current, the direction of current flow is continuously changing, as if the electrons can t make up their mind which way to go. It may help to think of a.c. current flow as a wave-like motion of the electrons. We know that waves will propagate through water without actually translating the water molecules over large distances. Mathematically the current flow can be described as I(t) = I sin(ωt). The alternating current in the field coil creates a time-varying magnetic field B(t) = B sin(ωt) that is intercepted by the pick-up coil. E(t) = NBA ω cos θ cos(ωt). (3)

3 3 Ignoring the minus sign, which is included to get the direction of the current flow correct, we can think of the induced voltage as having the form: E(t) = E cos(ωt) where E = NBA ω cos θ = NωΦ B. (4) You will measure the induced voltage E in the pick-up coil by connecting the ends of the wire to an oscilloscope. Apparatus The apparatus consists of the following: The field coil, signal generator, oscilloscope and pick-up coil should be attached together as shown in the schematic diagram below: Function Generator Field Coil Oscilloscope CH1 input pick-up coil Procedure Setting up the Signal Generator 1. Switch on the power (LINE ON) 2. Set the frequency of the alternating current in the field coil To do this:

4 4 (a) Depress the 10k button. (b) Set the frequency control to Set the waveform type to sinusoidal In the top right-hand corner there should be three buttons that represent three different kinds of waveforms. Depress the one marked with a sinusoidal wave. (i.e. ) 4. The Amplitude setting can be set anywhere in the range 60-80% of MAX. This can be increased later if necessary. It should not be! 5. Check that the DC OFFSET is set to zero. Setting up the Oscilloscope 1. Switch On by depressing the POWER button A green light above the power button should turn on, and a green line may show up on the display. 2. Place the pick-up coil in the center of the field coil, and position it so that is coaxial with the field coil, as shown below: 3. Set the SOURCE switch to INT. 4. Set the MODE (middle) dial to CH1. Only the signal received through the CH1 (i.e. Channel 1) INPUT is displayed. 5. Set MODE (upper right) switch to NORM

5 5 6. Adjust the vertical POSITION knob until you see a signal. 7. Adjust the INTENSITY and FOCUS dials to get the clearest, sharpest display image you can. 8. Adjust the TIME/DIV and CH1 VOLTS/DIV dials until you get a clear, stable signal that takes up as much of the screen as possible while still staying within the onscreen grid. Start with a TIM/DIV of 50 µs/div. Make a note of your TIME/DIV and VOLTS/DIV settings in your lab book. 9. Center the signal using the vertical and horizontal POSTION dials. 10. Tuning into the resonance frequency. You should now vary the frequency dial on the Function Generator to maximize the signal on the Oscilloscope. When the frequency gets close to the resonant frequency the signal on the oscilloscope display may exceed the vertical limits of the display. You should adjust the VOLTS/DIV knob to bring the entire signal back onto the display. Your display should now look something like this: Measuring Voltage with the Oscilloscope Throughout this lab you will have to make several measurements of the voltage across the pick-up coil using the oscilloscope. 1. Using the vertical POSITION knob move the sine wave vertically, so that the minimum just touches the bottom line on the display.

6 6 2. Now use the horizontal POSITION knob to move the sine wave horizontally until a maximum lines up with the ruled line in the center of the display. V pp For the purposes of data collection and analysis, we will use the peak-to-peak voltage V pp = 2E. Always be sure to record the error associated with any measurements you make. PART A: Measuring the Dependence of V pp on x In this part of the experiment you will measure the dependence of the magnetic field strength B(x) on the distance x from the center of the coil along an axis through the coil center and perpendicular to its plane.

7 7 As derived in Giancoli Example 28-10, the magnetic field for a circular coil as a function of x is: B(x) = µ 0INr 2, (5) 2(r 2 + x 2 ) 3/2 Where µ 0 is the permeability of free space, I is the current flowing through the field coil, r is the radius of the field coil, and N is the number of turns in the field coil. To measure V pp as a function of x: 1. Insert the longer of the two Teflon rods into support block and secure it in place. 2. Slide the pick-up coil onto the rod. Position it so that V pp is at a maximum (i.e. make the sin wave on the oscilloscope s display as tall as possible). This should happen at the very center of the field coil. Take x to be 0 at this point. This should be done by holding the wires rather than the coil because the presence of your hand will modify the pickup signal. 3. Measure and record both V pp and x. 4. Slide the pick-up coil 2.0 cm down the shaft. Measure and record both V pp and x. 5. Repeat step 5 until you reach x = 22 cm. PART B: Measuring the Dependence of V pp on θ Previously we found that V pp = 2NBAω cos θ, (6) where the cos θ term came from the flux Φ B = BA cos θ. By rotating the pickup coil relative to the field coil we can test this relationship. This will allow us to verify that the voltage induced in the pick-up coil depends upon the angle between the pick-up coil and the magnetic field generated by the field coil. To summarize, we are going to investigate the theoretical prediction that: V pp cos θ. (7) To measure V pp as a function of θ:

8 8 1. Insert the shorter of the two Teflon rods in the support block and secure it in place. 2. Slide the pick-up coil onto the rod and position it so that V pp is at a maximum. 3. Measure and record V pp. 4. Rotate the base 10, while being careful to make sure the pick-up coil does not move on its rod. Measure and record both V pp and θ. 5. Repeat step 5 until the arrow on the lid points to the 90 mark.

9 9 Name: Student ID Partners Name Date WORKSHEET: To be handed in: 1. Record the Radius of the Field Coil: VALUE ERROR Radius of Field Coil 2. PART A: Measuring the Dependence of V pp on x. (a) Data: x /cm x/cm V pp (x) V pp (x) V pp (x)/v pp (0) (V pp (x)/v pp (0))

10 10 (b) Plot of Theory and Experiment: Plot V pp (x)/v pp (0) versus x. On the same graph, plot the theoretical curve: b(x) B(x) B(0) = r 3 (r 2 + x 2 ) 3/2 (8) Verify, using the equations given, that this is the correct form for the theoretical curve. (c) Plot the following Difference Curve between Experiment and Theory: V pp (x) b(x) (9) V pp (0) including error bars in the experimental points. If the error bars display a trend (e.g. they get bigger or smaller), explain it. 3. PART B: Measuring the Dependence of V pp on θ (a) Data: θ θ V pp (θ) V pp (θ) V pp (θ)/v pp (0) (V pp (θ)/v pp (0)) (b) Plot of Experiment and Theory: Plot V pp (θ)/v pp (0) versus θ and include your calculated error bars on the experimental points. On the same graph plot the function cos θ

11 11 (c) Plot the following Difference Curve between Experiment and Theory: V pp (θ) cos(θ) (10) V pp (0) including error bars in the experimental points. If the error bars display a trend (e.g. they get bigger or smaller), explain it.