Experiment 10: Helmholtz Coils

Transcription

1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02 Fall 2003 OBJECTIVES Experiment 10: Helmholtz Coils To measure the magnetic fields of the following configurations: 1. one coil with N turns carrying curret 2. two coils with N turns each carrying currents in the same direction 3. two coils with N turns each carrying currents in opposite directions INTRODUCTION Consider the Helmholtz Coil Apparatus shown in Figure The Apparatus consists of two coils that are separated by a distance equal to their common radii. Magnetic Field of an N-turn Coil Figure 10.1 Helmholtz Coil Apparatus For a single coil of radius R with N turns carrying current I, the magnetic field due to the coil at a distance x along the axis passing through the center of the coil and perpendicular to its plane can be calculated using the Biot-Savart Law (see the 8.02T Study Guide, Worked Example for a derivation). The result (plotted in Figure 10.2) is 2 Nµ 0I R 1 B= 2 ( x + R ) 2 2 3/2 xˆ. (10.1) E10-1

2 Figure 10.2 The magnetic field of a single coil of wire along its axis. Prediction 1 (answer on the tear-sheet at the end!!): From our expression in equation Nµ 0I (10.1), we have the field at the center of the coil B ˆ center = x. Our coils have 2R 168 turns and a radius of R = 7.0 cm. If we run 0.6 amps through the coil, what is the magnetic field at the center of the coil in Tesla? In Gauss? 4 7 ( 1gauss = 10 Tesla, µ = 4π 10 ). o Figure 10.2 shows the magnitude of the field along the axis of the coil. What about the shapes of the field lines off-axis? The shapes of the magnetic field lines for a single coil of wire are shown in Figure The field directions shown are appropriate for current in the coil running counter clockwise when viewed from above. Another way to describe this is if you put the thumb of your right hand vertical, then your fingers will curl in the direction of the current flow. Figure 10.3 The magnetic field lines of a single coil of wire. E10-2

3 Prediction 2 (answer on the tear-sheet at the end!!): Suppose you move from left to right along the horizontal path indicated in Figure 10.3 above. Predict the behavior of the x-component (i.e. the vertical component) of the magnetic field as you move along that path, and draw it in the panel of Figure 10.4 and on the tear-sheet at the end. Figure 10.4 Your prediction of the behavior of the x-component (i.e. the vertical component) of the magnetic field as you move along the path shown in Figure 10.3 Prediction 3 (answer on the tear-sheet at the end!!): Suppose you move from left to right along the horizontal path indicated in Figure 10. Predict the behavior of the z- component (i.e. the horizontal component) of the magnetic field as you move along that path, and draw it in the panel on Figure Figure 10.5 Your prediction of the behavior of the z-component (i.e. the horizontal component) of the magnetic field as you move along that path shown in Figure 10.3 E10-3

4 Magnetic Field of a Helmholtz Coil The Helmholtz coil consists of two identical coils with the same axis, separated by a distance along their common axis equal to their common radii. When the current through both coils is in the same direction, the magnetic field at a distance x from the midpoint between the two coils is given by the sum of two equations in the form of Equation (10.1), suitably displaced from zero: 2 2 Nµ 0I R 1 Nµ 0I R 1 B ˆ ˆ H = x+ x. (10.2) 2 3/2 2 3/ ( x R/2 ) + R ( x + R/2) + R Near the midpoint ( x << R ), the above equation can be expanded in a Taylor Series as: B Nµ 0I R 6 x 6 x ˆ ˆ H( x ) = 1 2 3/ = c (5R 4) 5 R x B 5 R x (10.3) where B Nµ I R 2 c 0 = 2 3/2 (5R 4) xˆ (10.4) is the value of the magnetic field at the center of the coil ( x = 0 ). Notice that in this limit ( x R ), the field between the coils is nearly constant. We use this configuration of coils for precisely this reason it gives us a region where the magnetic field is reasonably constant near the value given in (10.4). The magnetic field along the common axis of the two coils (that is, a plot of equation (10.2), is given on the next page in Figure 10.3 (see the plot labeled Helmholtz Coil Configuration, meaning the current in the coils flows in the same direction). Magnetic Field of Two Coils With Helmholtz Spacing but Opposite Currents When the currents through the coils are in the opposite directions, the magnetic field near the midpoint between the coils is nearly zero and is given by the difference of two equations in the form of Equation (10.1), appropriately displaced; that is equation (10.2) with a minus before one of the terms instead of a plus. As we mentioned above, the above calculations indicate that when current flows in the same direction in the two coils, the resulting magnetic field is nearly uniform in the region near the midpoint between the coils. On the other hand, when currents flow in the opposite direction in the two coils, the magnetic field is close to zero in that region. E10-4

5 Figure 10.3 shows plots of the three cases discussed above, assuming that the current I through the coils is always the same magnitude. Figure 10.6: Plots of the magnet field on the axis of the coils discussed above APPARATUS Each coil in our Helmholtz Apparatus consists of 168 turns. The two coils are a distance R = 7.0 cm apart, a value equal to their mean radii. The resistance in each coil is approximately 2.8 Ω. To measure the magnetic field of these coils we will use the magnetic field sensor connected to the 750 Interface. An adjustable DC Power Supply will provide the current. SETUP A. Connecting The Magnetic Field Sensor To The 750 Interface The magnetic field sensor is plugged into the first analog channel A of the 750 Interface, as in previous experiments. Figure 10.7 The top of the magnetic field sensor, showing (from right to left) the RANGE SELECT switch, the TARE button, and the RADIAL/AXIAL switch, which is set to RADIAL E10-5

6 Remember that the Magnetic Field sensor measures one component of the magnetic field at a time. When the switch is set to AXIAL, the sensor will measure the component of the magnetic field along the axis of the probe, and will give a positive value when the magnetic field is pointing into the white dot on the very end of the probe. When the switch is set to RADIAL, the sensor gives a positive measurement when there is a component of the field perpendicular to the flat surface of the probe oriented into the surface of the probe at the white dot. We will be using both the AXIAL and the RADIAL settings in this experiment. Before you begin the experiment below, make sure the RANGE SELECT switch on the magnetic field sensor is set to 10, and the RADIAL/AXIAL switch is set to RADIAL. Download the Data Studio file exp10.ds from the web page and save it on your desktop. If there is already a file by this name on your desktop, save over it, as it may not be set up properly. Open the file by double clicking on it. Your file has a Graph Display that is already set up to display Magnetic Field Strength versus time. EXPERIMENT Magnetic field of a Single Coil with N Turns Connect the upper coil of the Helmholtz coil (Figure 10.1) to the DC power supply so that current flows only through the upper coil. Adjust the power supply so that the voltage is about 1.5 V and the current is about 0.6 A (these do not have to be precise) 1. Use the AXIAL setting of your Magnetic Probe Sensor (commonly known as a Hall Probe ) to measure the magnetic field along the axis of the coils. Start the probe at the bottom of the Helmholtz coil (in the wooden hole there) and move the probe upward as you take data. Start taking data by clicking the Start button. Try to move the probe upward with as uniform a velocity as possible, moving the probe as far above the upper coil as you started below the upper coil (roughly the coil radius). Your results will be displayed on the DataStudio graph. You may find that it s hard to do all that s needed simultaneously. Try having one group member start the data sampling while another moves the probe. After some practice, you might get a smoother graph. Question 1 (answer on the tear-sheet at the end!!): Does this graph look like the graph in Figure 10.2? Question 2 (answer on the tear-sheet at the end!!): Does this maximum value in gauss of your measured field agree qualitatively with your Prediction 1 above? E10-6

7 2. Use the RADIAL setting of your Hall (Magnetic) Probe Sensor to measure the vertical component of magnetic field as you sweep across the path above the single coil (passing through the axis of the coil) shown in Figure Which way should the white dot on your sensor be facing for an upward magnetic field to show up as positive in your measurement? Question 3 (answer on the tear-sheet at the end!!): Does this graph look like the your predicted curve in Figure 10.4? 3. Use the RADIAL setting of your Hall (Magnetic) Probe Sensor to measure the horizontal component of magnetic field as you sweep across the path above the single top coil (passing through the axis of the coil). Again, which way should the white dot be facing? Question 4 (answer on the tear-sheet at the end!!): Does this graph look like your predicted curve in Figure 10.5? Magnetic field of Two N-turn Coils in Helmholtz Configuration Connect the upper and lower coil to the DC power supply so that the coils are connected in series with the current flowing in the same direction through the coils. This will mean connecting the red post from one coil to the black post of the other. Adjust the power supply so that the voltage is 1.5 V and the current is approximately 0.30 A. Note that this is now half the current as in the first part (part A) of this experiment (why is this?), so to compare with the graphs in Figure 10.6, which always assume the same current, you must make appropriate allowances. 4. Use the AXIAL setting of your Hall (Magnetic) Probe Sensor to measure the magnetic field along the axis of the coils. Repeat the same procedure as in (1) above, page E10-6. Question 5 (answer on the tear-sheet at the end!!): Does this graph look like the appropriate graph in Figure 10.6? Magnetic field of Two N-turn Coils with Currents in Opposite Directions Connect the upper and lower coil to the power supply so that the coils are connected in series with the current flowing in opposite directions through each coil. Adjust the power supply so that the voltage is 1.5 V and the current is approximately 0.30 A. 5. Use the AXIAL setting of your Hall (Magnetic) Probe Sensor to measure the magnetic field along the axis of the coils. Repeat the same procedure as in (1) above, page E10-6. Question 6 (answer on the tear-sheet at the end!!): Does this graph look like the appropriate graph in Figure 10.6? E10-7

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9 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Experimental Summary 11: Helmholtz Coils Group Names Magnetic Field of a Single Coil Prediction 1: If we run 0.6 amps through a single coil, what will be the magnetic field at the center of the coil in Tesla? In Gauss? Tesla Gauss Prediction 2: Predict the behavior of the x-component of the magnetic field as you move along the path in Figure Prediction 3: Predict the behavior of the z-component of the magnetic field as you move along the path in Figure E10-9

10 Figure 10.6: Plots of the magnet field on the axis of the coils discussed above Question 1: Does this graph look like the graph in Figure 10.2 or the appropriate graph in Figure 10.6 above? Question 2: Does this maximum value in gauss of your measured field agree qualitatively with your Prediction 1 above? Question 3: Does this graph look like the your predicted curve in Figure 10.4, e.g. your Prediction 2 above? Question 4: Does this graph look like the your predicted curve in Figure 10.5? Question 5: Does this graph look like the appropriate graph in Figure 10.6? Magnetic field of Two N-turn Coils with Currents in Opposite Directions Question 6: Does this graph look like the appropriate graph in Figure 10.6? E10-10