MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics 8.02

Fall 2003

Experiment 10: Helmholtz Coils OBJECTIVES 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 10.1. The Apparatus consists of two coils that are separated by a distance equal to their common radii.

Figure 10.1 Helmholtz Coil Apparatus Magnetic Field of an N-turn Coil 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 9.10.7 for a derivation). The result (plotted in Figure 10.2) is G N µ0 I R 2 1 B= xˆ . 2 2 ( x + R 2 )3/ 2

(10.1)

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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 G N µ0 I (10.1), we have the field at the center of the coil B center = xˆ . 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? ( 1gauss = 10−4 Tesla , µo = 4π × 10−7 ).

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 10.3. 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.

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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 zcomponent (i.e. the horizontal component) of the magnetic field as you move along that path, and draw it in the panel on Figure 10.5.

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

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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: G N µ0 I R 2 N µ0 I R 2 1 1 ˆ + BH = x xˆ . (10.2) 3/ 2 3/ 2 2 2 ( x′ − R / 2 ) 2 + R 2   ( x′ + R / 2 ) 2 + R 2      Near the midpoint ( x′