Magnetic Force and the Determination of µ Physics 208 Purpose: To determine the permanent magnetic dipole moment,µ, using Helmholtz coils and a small neodymium-ironboron permanent magnet. Introduction: We will investigate the force exerted on a permanent magnet in the center of two coils carrying the same magnitude of current. These two coils have a radius equal to their separation. This special coil configuration are called Helmholtz coils, named after the German scientist and philosopher who made fundamental contributions to physics. The particular coil we are using has a radius value of 7.0 cm and 168 turns. It turns out that a current carrying coil behaves as a magnetic dipole. Figure 1: Coil as A Magnetic Dipole. The magnetic field at point z along the axis of N turns of wire in the coil all in close proximity and all carrying the same current, I is: B z = µ oin 2 R 2 [z 2 + R 2 ] 3 2 (1) Figure 2: Single Coil. 1
Now, if we consider the magnetic field generated by two identical coils of N turns, radius R, and the same current,i, passing through both coils, the expression for the magnetic field along the z-axis as a function of current at the exact center of the set of coils is: B z z =.859µ on I R 2 (2) The force produced by this axial magnetic field is: F z = µ B z z (3) B z z is called a magnetic gradient which is analogous to the temperature gradient defined for thermal conductivity in your text book. The magnetic moment is a property of the magnet. Both the magnetic field and magnetic moment are vectors having a magnitude and direction. In this lab, we will quantitatively show there is no net magnetic force on a permanent magnetic dipole if the current in the coils are in the same direction producing a constant magnetic field. Therefore, a net magnetic force on a magnetic dipole exists only when it is in the presence of a magnetic field that varies in space. Once that varying magnetic field is established we can determine the magnitude of the permanent magnetic moment of the dipole. Figure 3: Experimental Apparatus Laboratory Procedure: Part I.I - Taking Direct Measurements: Calibration - Hooke s Law 1. Place the magnet which is mounted in the plastic gimbal inside the tower such that the bottom of the gimbal can be read on the scale. This is your equilibrium position. 2. We use gravity to exert a force on the spring. Be careful NOT to overstretch the spring. Use the five 1g steel ball bearings as masses (assume these values are exact with no uncertainty in the mass). Add the masses to determine the spring elongation for 1, 2, 3, 4, and 5 ball bearings and record the displacement z for each mass. 3. Determine δ z from the precision of the scale attached to the apparatus. 2
4. Determine the fractional uncertainty (δ z/ z) for this measurement and record this in your data 5. Calculate the Force F by converting the masses into kilograms and multiplying by the acceleration due to gravity (g = 9.81 m/s 2 ) 6. Draw a full page graph of F (y axis) vs z (x axis) and determine the spring constant (k) from the slope. (F = k z) 7. The uncertainty in the slope δk is equal to δ z so determine and record the fractional uncertainty for the spring constant (δk/k). Figure 4: Magnetic Field of Helmholtz Coils with Same Direction of Current in Both Coils. Part I.II - Taking Direct Measurements: Uniform Magnetic Field 1. Connect the two coil system in series such that the current flows in the SAME direction in each coil. SEE FIGURE 5 on next page for help. Have your TA check your wiring. 2. Adjust the brass rod so the magnetic dipole hangs about 1-2 cm above the center of the two coils. 3. Connect the coils to the power supply and slowly turn up the voltage until you read.5 amperes of current. 4. Determine δi from the precision of the DVM. 3
Figure 5: Experimental Apparatus Wiring 5. Determine the fractional uncertainty (δi/i) for this measurement and record this in your data 6. Record the deflection, z. 7. Determine δ z from the precision of the scale attached to the apparatus. 8. Determine the fractional uncertainty (δ z/ z) for this measurement and record this in your data 9. Repeat this procedure for 1, 1.5, 2 and 2.5 amperes. 10. Draw a full page graph of z vs I. Part I.III - Taking Direct Measurements: Spatially Varying Magnetic Field 1. Connect the two coil system in series such that the current flows in the OPPOSITE direction in each coil. See Figure 6. Have your TA check your wiring. 2. Adjust the brass rod so the magnetic dipole hangs about 1-2 cm above the center of the two coils, near zero on the scale. 3. Connect the coils to the power supply and slowly turn up the voltage until you read.5 amperes of current. 4. Determine δi from the precision of the DVM. 5. Determine the fractional uncertainty (δi/i) for this measurement and record this in your data 6. Record the deflection, z. 7. Determine δ z from the precision of the scale attached to the apparatus. 8. Determine the fractional uncertainty (δ z/ z) for this measurement and record this in your data 4
(a) Uniform (b) Varying Figure 6: Uniform and Varying Magnetic Field Connections 9. Repeat this procedure for 1, 1.5, 2 and 2.5 amperes. 10. Draw a full page graph of z vs I and determine the slope. 11. Calculate µ based on equation #3 for force due to a magnetic field. z = µ.859µo N I kr2 (4) Part II - Determining Uncertainties in Your Final Values In the results section of your notebook, state the result of part I.III of your experiment in the form µ±δµ. Note, δµ in your measurements should be equal to the largest fractional uncertainty from your values of deflection z or current I fractional uncertainties multiplied by your value of µ. Example; δ z δi, δµ = µ max z I You should also address the following questions for parts I.II and I.III: 1. What can you infer from your plot in the Uniform Magnetic Field section? 2. You should compare the value of µ to its expected value of 4 A-m2. Does it fall within the experimental uncertainty? Which of the quantities measured has the most effect on the final uncertainty? If your results indicate that systematic or random error(s) may be present, try to determine some possible sources of the error in the experiment. 5