Electric and Magnetic Field and the Hall Effect In this lab you will interface a 3D Hall Probe to your microcontroller, and you will use the resulting “wireless B‐field probe” to do some interesting experiments. The Hall Effect comes about as a result of both magnetic electric forces acting on the current in a conductor. Charges moving in a magnetic field experience a Lorentz Force whose direction is perpendicular to both the velocity of the charge and the direction of the magnetic field: G
G G
F = qv × B . Consider a current I flowing in a conductor in the presence of a perpendicular magnetic field as shown below. Suppose each of the charges making up the current carries a charge +q. The charges will experience a Lorentz force which will drive them to one edge of the conductor, making that edge positive and leaving the opposite edge with a net negative charge. This, in turn, results in an electric field whose force on the charges will act in the opposite direction to the Lorentz force. An equilibrium will be reached when these forces are equal, and it is straight‐forward to show that the potential across the conductor is given by the formula IB
nqd
VH =
where n is the number density of charges and d is the thickness of the conductor. 1
• See if you can derive this expression for the Hall voltage. The only information you need in order to do this are the E and B force equations for the charge, and the assumption that all of the separated charges are located on the edges of the conductor as indicated by the + (blue) and – (red) areas in the diagram shown below. • By examining the equation for VH, can you see why semiconductors are usually used as Hall Probe elements rather than normal metal conductors such as copper? The formula for Hall voltage tells us that we can use this technique to measure magnetic fields if the other quantities are known, or can be calibrated. This is precisely the plan for this lab. 2
Interesting Aside: In considering most circuits it makes no difference whether we say negative charges are flowing one way or positive charges are flowing the other. For example, we usually say that positive current flows from the positive side of a battery, through the circuit, then back into the negative side of the battery. This works for analyzing all circuits even though it’s wrong – we actually know that negative current (electrons) flow from the negative side of the battery, through the circuit, and back into the positive side. One fascinating thing about the Hall Effect is that it does distinguish between positive and negative charge carriers in a conductor – you get different answers if you assume positive current flows one way or negative current flows the other (see below). This lack of symmetry makes the Hall Effect a very important research tool. 3
Part 1: Using your Hall Probe The modern electronics industry is now able to make small and inexpensive semiconductor Hall probes for commodity sales, and we will use a device called the Ametes MFS‐3A which is composed of three Hall effect sensors and can measure all three components of a magnetic field at once (much in the same way that the accelerometer measured all three components of acceleration at once). GND +5
Z
X Y We have attached this device to a second microcontroller board in exactly the same way that we attached the accelerometer – using double sided tape and wire wrap. The axes are also the same. P2 P4 P6 P8
P1 P3 P5 P7
Microcontroller Signal P1 (ground) P2 (Vbattery) P5 (ADC input 2) P6 (ADC input 3) P7 (ADC input 4) GND to accelerometer Power to Hall probe1 X axis Hall probe output Y axis Hall probe output Z axis Hall probe output 4
To power the B‐field sensor you will need to use a different battery back. The reason for this is simple: The ideal operating voltage for the Hall Probe is around 5V, and the battery holder that comes with the micro holds 2 x 1.5V AAA batteries, which gives us only 3V. The 3 x AAA battery holder you will be given puts out 4.5V, which is close enough to 5V that the Hall Probe works well, and close enough to 3V that the micro also works well. First plug the AP into the USB port, and then power up the ED by plugging it into the power carrier and turning on the switch on the battery pack. Be careful when you mate the power carrier with the ED to make sure that they are properly aligned as shown above. Now run the “Fields.exe” application: EnLiST_Ccode\executables\Fields.exe This time you will select “Hall Probe Mode” from the pull‐down Mode menu. If the probe does not appear to be acquiring data after doing this, close the program, unplug both the ED and the AP, and start over.
5
Just as you did with the accelerometer, the first time you run this program for the Hall probe you will need to do a quick calibration. The program will again ask you to hold the probe in different orientations, which should be easy since the axes are defined to be the same relative to the face of the probe circuit board. After the calibration, the x, y and z components of the B field shown will be defined relative to the total strength of the Earth’s magnetic at your location Be. In other words, if the probe is oriented so that its x axis points in the same direction as the Earth’s magnetic field, you will see Bx/Be=1, By/Be=0, and Bz/Be=0, Select “Show Plot” to see a plot of the three B field components and their evolution with time. Move the probe around and see what happens. 6
Use your probe to measure the direction of the magnetic field of the earth. Stay clear of metallic objects as these may perturb the field. The field components shown by the program are normalized to the magnitude of the total Earth’s magnetic field B0 where you did the last calibration, so Be2 = Bx2 + By2 +Bz2 should be very close to 1. (The magnitude of the Earth’s magnetic field in this part of the world is actually about 50μT.) • Start by orienting the probe so that Bx is about +1 and By and Bz are both close to zero. In this orientation the x axis of the probe is pointing in the same direction as the magnetic field of the Earth: • Is the direction of the earth’s field consistent with your expectations? • Now hold the probe so that the y axis points up and the z axis points toward the north (north is the direction toward the back of the room). Notice the relative size and signs of Bx, By and Bz. Make sure you understand how this consistent with the first measurement you made. • See if you can make a model explaining the earths magnetic field involving a big bar magnet inside the earth. 7
You will now investigate the magnetic field from a straight wire carrying a current. Tape a wire to the surface of the lab bench and put your Hall Probe on top of it. Measure the magnitude and direction of the field with the power supply off and with the power supply at maximum current (about 3 amps). • Use the above measurements to calculate magnitude and direction of the magnetic field due to the current in the wire in units of B0. Verify that this is proportional to the current in the wire. • How should you orient the wire so that you can measure the field from the current without having to subtract out the magnetic field of the Earth? • The magnitude of the magnetic field a distance R from a wire carrying current I is given by μ0I/2πR where μ0 = 4πx10‐7 Tm/A. Use this to figure out the actual distance R from the center of the wire to the sensing element inside the blue case. • Measure the magnitude if the magnetic field produced by the wire as a function of R. You will now investigate the field of a small neodymium magnet. • Use your Hall probe to find the north pole of the magnet. • Alight the magnet so that its north pole is facing your Hall probe and measure the magnitude of the field from the magnet as a function of distance as you move the magnet further away in small steps. You can start with the magnet quite close to the probe. • Optional question: These small magnets look very much like ideal magnetic dipoles. Given that the field like from a magnetic dipole look the same as the field lines from an electric dipole, come up with a conceptual argument why the field diminishes with distance faster that 1/R2 (you should have measured something like 1/R3 far from the little magnet). 8
Electromagnetic Waves The term “electromagnetic waves” may sound complicated, but in reality it is just light – light in any of its many forms. We encounter different types of electromagnetic (or EM) waves every day. X‐rays are used all the time in medicine, we use microwaves to cook our food, and ultraviolet rays from the sun give us sunburns. Radio waves carry music and news, and infrared signals from a TV remote tell the TV to change the channel. We’ve seen how charges and currents give rise to electric and magnetic fields. It’s not a large jump in logic to see that if we move our charges or change our currents, these fields must also change. As we accelerate our charges, we cause changes or “kinks” in the field. These changes propagate outward as EM radiation – in other words, light. 9
Look at how moving charges cause changes in electromagnetic fields. • Go to http://www.colorado.edu/physics/2000/waves_particles/wavpart4.html • Wiggle the negative charge around. Look at what happens to the electric field, and then how the positive charge responds. Note that when the negative charge is wiggled around, it creates a “kink” in the electric field. This kink travels outward at the speed of light. When the kink gets to the positive charge, that charge is accelerated. • Go to http://www.cco.caltech.edu/~phys1/java/phys1/MovingCharge/MovingCharge.html • Select “Linear” and “v=0” o How do the field lines spread out? • Try this with v set to 0.5, 0.75, 0.95. o How do the field lines change? • Try “Circular” with “v=0.5” o Look at how waves in the electric field are radiated. • Select “Linear” and slide the “v” slider back and forth between “v=0” and “v=0.95” several times. o How do the waves move out? In what directions are the radiated waves maximized? In what directions are there no waves radiated? This is basically how light is radiated. Accelerating charges produce radiation, and radiation accelerates charges. These radiated “kinks” are nothing more than light. These are really the basics. However, there are a couple of details that we need to cover. 10
Maxwell’s Equations The mathematical equations govern the behavior of electric and magnetic fields are called Maxwell’s Equations. (1)
(3)
(2)
(4)
Fully interpreting these equations can be a lifetime of work, but the basic ideas are very simple: Equation 1 says that electric fields are produced by charges. Equation 2 says that there are no magnetic charges – that you can’t have a north without a south. Equation 3 says that a changing magnetic field also produces an electric field (also known as Faraday’s Law), and Equation 4 says that currents produce magnetic fields and that changing electric fields also produce magnetic fields. The fact that changing electric fields produce magnetic fields and changing magnetic fields produce electric fields is ying‐yang that allows light waves to propagate. The accelerating charge and its changing electric field – the “kink” we saw earlier – also produces a magnetic field. This magnetic field changes along with the changing electric field and they propagate together as an electromagnetic wave. This process of a changing magnetic field causing an electric field, described by Faraday’s Law, is called induction. Since it is such an important part of electromagnetism, we will investigate induction in more detail in the next few activities. 11
Investigating electromagnetic induction. You are going to investigate the effect of a changing magnetic flux on a loop of wire, which is probing Faraday’s Law (the top right Maxwell equation on the previous page). To do this you will need a loop of wire. Using the roll of magnet wire provided, wind a loop of a couple of dozen turns loosely on your finger, leaving at least 6 inches free at each end. Pry it off your finger and scrape about ¼” of insulation off both ends of the wire and your loop is ready. Attach one end of the coil to the ground pin on your ED (Pin 12) and the other to the Vin1 input (Pin 3) as shown in the diagram to the right. The signals you will be observing are small so you will need to tell the ED to apply the biggest gain it can, which is about a factor or 220. This is done using the “Mode” pull‐down menu as shown. With this setting the blue trace (and Vin2 reading) is about 220 times the red trace (Vin1 reading). In this mode Vin2 is generated internally by the microprocessor based on Vin1 , and any signal present on Pin 4 is ignored. 12
Take a strong magnet and wave it back and forth close to the coil – you should see something like the following picture on the screen: According to Faraday’s Law, changing the magnetic flux through the area of a loop induces an electric field in the loop itself. This field will show up as a voltage across the ends of the wire that makes up your loop, the sign of the voltage indicating the direction of the induced current. • Knowing this, what’s wrong with the above picture? Why is the voltage never negative? In order to be able to see negative voltage swings in the loop, we will clearly need to hook it up differently. The following circuit will do the trick. Hook it up using your breadboard as shown. 13
Wave the magnet back and forth close to the coil – you should now see something like this: Much better – if we assume the initial horizontal line represents the condition where no current is induced in the loop, then we are now able to see both positive and negative currents being induced when we move the magnet around. • Discuss how this simple little circuit works. Find a way to figure out which side of your magnet is its north pole. Hold the magnet so that its North is pointing down and “fly” it back and forth over the coil. • Explain why you get qualitatively the same shaped curve whether you are moving the magnet left‐to‐right or right‐to‐left. • Try moving the magnet faster and slower as you fly it past the coil and explain your observations using Faradays Law. • Try turning the magnet over so the south pole faces down as you fly it past the coil and explain your observations using Faradays Law. • Try turning the magnet sideways so that the N‐S axis is parallel to the velocity of the magnet as you fly it past the coil and explain your observations using Faradays Law. Spend some time doing the above activities and make sure that it all makes sense to you before moving on. 14
A look at transformers. We have just seen that a changing magnetic field causes a voltage to be induced in a coil of wire. We have also showed that a current in a wire produces a magnetic field. Both of these facts are used to explain how a transformer works: A changing current in one coil of wire (called the “primary”) causes a changing magnetic field that induces a current in a second coil of wire (called the “secondary”). We will be using a transformer to “step up” (i.e. increase) the signal in the next couple of activities, so before doing that lets do a few simple activities to get familiar with transformers. • Connect the input of the transformer (the red and white wires) to the function generator on the oscilloscope. Input a sine wave of about 1 V amplitude, and 1 kHz frequency. The setup should look something like the picture on the next page – your instructor can help you set it up if you are still a bit rusty using oscilloscopes: 15
• What does the transformer do to the input voltage? Can you explain what’s going in terms of the fact that the transformer you are using has 10 times as many loops of wire on its secondary side (blue‐green) as on its primary side (red‐white). • Predict what you would see if you turn the transformer around and attach the secondary side (blue‐green) to the signal generator and look at the signal on the primary side (red‐white). Try it and wee if you were right. 16
Sending Information by EM Waves We not only want to understand how EM waves work, we also want to be able to use them. In our case, we specifically want to be able to use them for communication. If we were to try to send information via a pure EM wave, we wouldn’t be able to send much. A pure sine wave is characterized by its amplitude, frequency, and phase. The amplitude naturally dies off with distance, so different observers would measure different values. The phase depends on what time you call t = 0, so again, that would not be useful for sending a message. Only the frequency would be able to communicate any information. In stead of sending messages in the wave itself, we send messages encoded in small changes to the wave. This is called modulating the original “carrier” wave. We can modulate the amplitude of the wave. This is how AM (Amplitude Modulation) radio works. An extreme example of this would be Morse Code. One can see that a lot of information can be encoded in this way. 17
We can also modulate the frequency of the wave. This is how FM (Frequency Modulation) radio works. We could also modulate the phase, but this turns out to be basically the same as frequency modulation. (This is why we don’t have “Phase Modulation” radio.) Building a laser radio. We are going to use a laser pointer to send sound along a light wave. Our laser radio consists of two parts, a transmitter and a receiver. We will use a transformer to modulate the voltage that powers the laser pointer. This will modulate the brightness (i.e. amplitude modulation) of the laser beam. We will be able to detect these changes in brightness with a photo‐resistor across the room and then change this back into sound. Transmitter Receiver • Build the laser receiver and transmitter as shown. (The black wires leading out of view in the photos connect to an audio plug.) • Connect the audio output plug to one of the mini audio amplifiers / speakers. • Shine the laser from the laser pointer directly into the photoresistor on the receiver. 18
• In stead of the audio input wires, try connecting the input of the transformer to a sine wave from the function generator from the oscilloscope. Be sure to keep the amplitude of the input small enough that we don’t ruin the laser pointer. (Start with amplitude set to 0 and increase slowly.) (You could also connect the function generator directly to the output side, bypassing the transformer completely. See what works best for you.) o Try adjusting the frequency of the input. (Start around 400 – 500 Hz) How does that change the output? How does changing the input amplitude affect the output? (Remember, don’t change the input amplitude too much.) o Try slowing down the frequency to about 1 – 5 Hz. Can you see the brightness of the laser beam change? (It will be easier to see with some laser pointers than with others. It actually works better with cheaper laser pointers.) • Disconnect the function generator and connect the audio input wires. Plug the audio plug into a radio, computer speakers, or something like that. You can also use the microphone and the other audio amplifier to send your voice along the laser beam. o Try sending a signal along the laser beam. Can you pick it up at the other end? How good is the quality? Can you think of ways we could improve the quality? Can you send your message across the table? Across the room? Around a corner (with a mirror)? Try blocking the beam to see that the signal is really being carried by the beam. Note: When sending laser beams around the room, remember laser safety and be careful. In the last activity, we used visible light to carry a message. However, in this regard, there is nothing special about visible light. We could just as easily have used another part of the EM spectrum, like radio waves, as we’ll see next. 19
Building an AM radio transmitter. We will use a crystal oscillator to send an AM radio signal. Our crystal oscillator is a little device built so that when we put a voltage across two of the leads, the third lead switches voltage 1 million times per second, i.e. at f = 1 MHz. This frequency is right in the middle of the radio wave portion of the EM spectrum. We will again be using our transformer to modulate the amplitude of the resulting radio wave. • Connect the crystal oscillator to the battery directly without the transformer or other components. o Look at the voltage on the antenna lead with the oscilloscope. You’ll need the time resolution to be about 1 μs. • Build the radio transmitter as shown above. (The black wire leading off to the left in the photo is our audio input wire, and the black wire at the right is a simple antenna, which we will talk about later.) 20
• As before, connect the audio input plug to the computer speakers or something like that. o Again look at the voltage on the antenna lead with the oscilloscope. This time look at it both at the 1 μs time scale, and also at the ≈ 10 ms time scale. What do you see? Do you see how the input signal modulates the carrier wave? We can see the resulting wave both at the frequency resolution of the carrier wave (≈ 1 μs), and at the frequency resolution of the signal (a few milliseconds). Before connecting an antenna, we need to think a little about this. All an antenna is just a piece of wire. However, the length of the wire is important if you're going to be able to get a good signal. The ideal length for an antenna is 1/4 the wavelength of the EM wave. o Can you figure out the wavelength of our radio waves? What would our ideal antenna length be? Is this a reasonable (or legal) length for us to build? (But does it explain why a lot of the radio transmitter towers you see are so tall!) o Would an FM radio transmitter antenna be longer or shorter? By how much? (You might then ask why we aren't building an FM radio transmitter. The answer is that the internal electronics are a little more complicated and tricky to understand. You can do it, but you'll have to look this one up on your own.) Clearly, we can't really build an antenna long enough to be exactly right for our signal. There are various things we could do to make a short antenna (like 1–2 m long) transmit better like building a resonator, or one of a number of other strategies that people have come up with, but we really won't have time for this here. These are also things that you can look up on your own. For our purposes, the voltage probe on the oscilloscope will actually transmit a couple of inches. This is all we'll need. • Tune the portable radio to 1000 kHz AM. o Can you pick up the signal from your radio transmitter? (You'll have to hold the radio right up near the output of the crystal oscillator.) What kind of range does it have? Naturally, like we said, there are lots of things that you can do to improve how your transmitter works, but you really have the basics of a working AM radio station. 21