Faraday's Law

Faraday's Law - Electromagnetic Induction: Faraday's Law...

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Unformatted text preview: Electromagnetic Induction: Faraday's Law OBJECTIVE: To understand how changing magnetic fields can produce electric currents. To examine Lenz's Law and the derivative form of Faraday's Law. EQUIPMENT: Circular Coils apparatus, PC sound card, FFTScope, magnet, paper clip, cables INTRODUCTION: Have you ever wondered how a telephone works? You may recall (perhaps in middle school or high school) that sound waves from your voice are converted to electricity by the microphone and that electricity is converted back to sound waves by the speaker, but how does that actually happen? Such inventions as the telephone, electric generators, electric guitar pickups, electrical transformers, car cruise controls, induction stoves and blood flow meters all exploit the fact that a changing magnetic field can give rise to an electrical current, a phenomenon we call electromagnetic induction. The mathematical law that relates the changing magnetic field to the induced current (or, more accurately, the induced voltage or emf) is called Faraday's Law, named after the man (among others) who first observed it in the laboratory around 1830. You may recall from lecture that the magnetic flux through a surface (like the surface of a conducting loop) that is immersed in a magnetic field B is = B A cos where is the angle between the magnetic field and the direction perpendicular to the surface. A conducting loop which has an ammeter attached to it will register a current if the magnetic flux through the loop changes in time. One way to change the flux through a conducting loop is by moving a magnet in its vicinity. In the figure below, a the North pole of a magnet is moved towards a conducting loop. As the magnet approaches the loop, the magnetic field within the loop increases which means the magnetic flux also increases. The changing magnetic flux in the loop will induce a current in the direction shown in the figure. You could also move the North pole of the magnet away from the conducting loop which would decrease the flux through the loop. Either way, a current would be induced in the conducting loop because all that is needed is a change in the magnetic flux. The difference between moving the North pole of the magnet towards the loop and moving the North pole of the magnet away from the loop will manifest itself in the direction of the induced current in the loop. You will learn about this later when you explore Lenz's law. Furthermore, moving the South pole of the magnet towards (away) the loop will induce a current in the same direction as moving the North pole away (towards) the loop. You may also recall from lecture that current flowing through a loop of wire produces a magnetic field similar to that of a magnet, and we can exploit this fact to create a changing magnetic flux in a second conducting loop that is placed nearby. Consider the situation shown in the figure below: Initially Loop 1 is connected to a battery by a switch and the switch is initially open. Loop 2 is just a closed loop of wire that is near Loop 1. In (a) there is no current flowing in either loop because the switch is open in loop 1. When the switch is closed as shown in (b), the increasing current in loop 1 produces a changing magnetic field and thus a changing magnetic flux within loop 2. This changing magnetic flux will induce a current in loop 2 even though there is no battery directly connected to it! More precisely, Faraday noted that the emf induced in a loop is proportional to the rate of change of magnetic flux though it: =− N d dt where is the electromotive force induced (measured in volts) and N is the number of turns of the coil. Provided each turn of the coil is sized and oriented like the others, its contribution is simply additive; hence the coefficient N in front of the flux derivative. Notice the negative sign. Lenz's Law states that the induced emf (and current) will be in a direction such that the induced magnetic field opposes the original magnetic flux change. Keep in mind that the induced current will now produce an induced magnetic field. The direction of that magnetic field will be opposite to the direction the flux is changing. APPARATUS: Examine the apparatus. There are three sets of coils, two of which are fixed with respect to each other (wrapped in black tape) and which share the same number of turns and diameter. These two coils (total number of turns N=20) act in concert with each other, each producing a magnetic field in the same direction. These outer coils produce a fairly uniform magnetic field inside the apparatus – recall the Helmholtz Coils setup from the E&M Forces lab that generated a field that remained constant throughout the path of the electron beam. The third inner coil (total number of turns N=11) is more rectangular than circular, smaller than the other two, and can be rotated with respect to them using the knob on the side of the apparatus. There are protractor markings around the knob to measure the angle between inner and outer coils. The FFTScope software, which controls the PC's sound card, will act as a signal generator. It will be used to drive the outer coils with a periodic waveform that will produce a changing magnetic flux, inducing a current in the inner coil. You can also induce a current manually (signal generator off) by moving a small magnet close to the inner coil. In both cases, you will use FFTScope to examine the induced current in the inner coils. PROCEDURE: NOTE: DO NOT DISCONNECT OR CHANGE THE APPARATUS WIRING. PLEASE EXERCISE CARE WITH THE COIL APPARATUS. TURNING THE ANGLE KNOB TOO MUCH IN ONE DIRECTION MAY DAMAGE THE WIRES! A. Induction by moving a permanent magnet You will induce an emf in the small coil manually, creating a changing flux through the small coil by moving the magnet's magnetic field through it. 1. Examine the small magnet on your table. One side of the disk corresponds to magnetic North, the other to South. Draw a diagram of the magnet, sketching magnetic field lines. Choose North and South poles on the magnet arbitrarily, and put arrows on the lines corresponding to your choice. Remember that field lines represent lines of force; the more closely together the lines are drawn, the stronger the magnetic force is at that point. If you do not have a clue, pick up the paper clip with one hand, hold the magnet with the other hand, and try to get a rough idea of where around the magnet the clip feels the most force. Draw your diagram on the hand-in sheet. 2. Rotate the knob on the side of the apparatus so that the angle marker is set to 0 or 180o. This should orient the plane of the inner coil parallel to the table. 3. Open FFTScope, which is in the Lab Software folder on the Desktop. Below are important buttons you will use in the FFTScope toolbar: 4. Turn data acquisition ON by pressing the GO/STOP button in the toolbar. You should see the message “DATA acquisition ON” on the lower left-hand corner of the FFTScope window. Click the autoscale button once – you will see random noise because the scale has been minimize. Click on the EXPAND button twice to expand the scale – you will need some headroom in the graph since you are about to induce a signal, using the small magnet. 5. Pick up the small magnet by the sides and hold it such that the plane of the disk is horizontal. Now position it right above the inner coil of the apparatus, without touching. Pull the magnet quickly upwards, while watching the trace on the FFTScope screen. Does the emf pulse trace go up or down? ______________ Record in hand-in sheet. 6. Flip the magnet over such that the side you previously had up is now down. If you repeated Step 5, which way would expect the pulse to move? Discuss this with your lab partner before proceeding. Record in hand-in sheet. Repeat Step 5. Which way did the pulse actually move? Again, record in hand-in sheet. 7. Keeping the same magnet orientation in the previous step, try rotating the inner coil 180o and again pulling the magnet upwards from it. Which way does the pulse move now? Record in hand-in sheet. 8. Try to get a rough estimate of emf pulse height – count the number of horizontal grid lines (and fractions thereof) on the FFTScope screen the pulse moves away from the baseline when you pull the magnet away. What was the maximum pulse height (in grid lines) that you could momentarily generate on the inner coil? (Record) 9. Now orient the inner coil such that its plane is vertical, i.e. 90o with respect to the table, and plane of inner coils. Pull the magnet away from the coil again. What was the maximum pulse height (in grid lines) that you could momentarily generate on the inner coil? Record. If you moved the magnet the same way you did before, why is this pulse height different? ______________________________________________________________ In this section we have physically moved a magnet in the vicinity of a coil, inducing a current in it. In order to do this we have performed work on the magnet because we had to oppose a force in order to keep the magnet moving, even at constant velocity. What is the source of this force? According to Lenz's Law, the induced current produces an induced magnetic field, which is represented by a magnetic moment (denoted by m on the diagram below, which opposes the motion of the magnet. This magnetic moment can be thought of as "virtual magnet" whose poles either oppose or attract the real magnet, depending on the direction of motion. In either case, the direction of induced current can be identified using your right hand: point your extended thumb in the North direction of the magnetic moment; your fingers will curl in the direction the current is going through the loop. Doing work on a magnet to create a current is the basis behind an electrical generator. In hydroelectric dams, falling water turns large paddles connected to electrical generators that convert mechanical energy to electrical energy. Conversely, we can use changing flux to do work. This is the principle behind an electric motor. B. Induction by varying the current You will now induce an emf in the smaller coil using a changing magnetic field supplied by the outer coils. There will be no motion of the coils involved. The signal generator will provide the outer coils with a fluctuating current to vary the magnetic field, and hence the magnetic flux, through the inner coils. The fluctuating current will be sinusoidal, triangular or square in waveform shape. 1. Predict what the response waveforms will look like for sine, triangle and square driving waves. Think of the mathematical relationship between magnetic flux (driving signal) and induced emf (response signal). Draw predicted waveforms in your hand-in sheet. 2. In FFTScope, go to Function Generator in the menu bar. It should be set to Silent. Change it to Sine. Click on the STEREO/MONO button to simultaneously view both driving waveform (blue) and induced waveform (red) the GO button. Press Autoscale to see the entire wave height. Click and drag a rectangle across a portion of the graph – this will “zoom in” on the waveforms, making them appear less compressed in the horizontal direction. You may have to Autoscale again after this. If you zoomed in too much, you can press the RESTORE button in the toolbar. Remember that you can always “freeze” the scope waveforms for closer inspection by clicking the GO/STOP button on the toolbar. Why doesn't the response waveform look “perfect”? There may be a significant distortion of the waveform due to the inductance of the two circuits (see the section on inductance your textbook). You may also notice some "squiggly" oscillations near the edge of the waveform, especially with the square wave – this is due to the slight capacitance in the coil (recall the oscillatory behavior of an RC circuit). 3. Switch to the other waveforms – triangle, square. Quickly check that the frequency and amplitude (driving signal only) remain the same as you toggle between them; only the shape changes. If you are seeing a noisy signal when you Autoscale, turn up the volume knob on the PC speakers; however, if you are getting flat-topped driving waveforms, you are overloading the sound card input - turn down the volume knob. 4. Sketch the induced response for the sine, square and triangular waveforms on the hand-in sheet. 5. Examine the graphs you just drew. What is the general relationship between the driving and induced waveforms? Write your answer on the hand-in sheet. Think of the mathematical relationship between magnetic flux and the emf and note that spikes in the graph represent a very rapid rate of change. 6. Let us now check how the magnetic flux through the smaller coil changes as we rotate it within the fairly uniform field of the outer coils. Increase the frequency to 5000 Hz by pressing F9 on keyboard and entering this value; this will give more sensitivity in the measurement of the response amplitude. Don't worry about the stability of the scope trace; we are only interested in examining the amplitude as a function of angle. Start off with an inner coil angle of 0o (coil plane parallel to table) – go up to View in the menu bar, and at the very bottom of the menu select “View peak-to-peak amplitude”. This will display two successive values: a number for R channel (driving) and a number for L channel (response). Record L, the peak-to peak amplitude. Vary the relative angle between inner and outer coils from 0o to 360o in 15o steps, each time recording the peak- to-peak amplitude. Record in hand-in sheet. You can plot Amplitude vs. Angle by printing out the graph page (last page), or you can plot in Graphical Analysis. Real-life Applications Faraday's Law is the basic principle behind the simple telephone. In a microphone there is a diaphragm, around which a coil is wrapped, which can move back and forth in response to sound waves. A stationary bar magnet, placed near the coil, induces current in the coil which can then be transmitted (with amplification) to the speaker of another telephone. Conversely, when the current reaches the speaker, which consists of another coil/diaphragm/magnet combination, the varying coil current causes the diaphragm to move and displace sound waves: Another direct application of Faraday's Law is a transformer. If the two sets of coils 1 and 2 similar to what you just used were of the same size, we can either increase or decrease voltage by varying the number of turns according to this equation: V 2=− N2 V N1 1 High-voltage transformers are used in conveying electricity from your electrical company to your home. Since power loss on the lines is equal to I2R, it makes sense to use a high voltage and low current when transporting power over great distances. Conversely, for safety reasons, low voltage and higher current is used in the home. ...
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This note was uploaded on 02/29/2012 for the course PHYS 227 taught by Professor Rabe during the Fall '08 term at Rutgers.

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