Magnetic Force: The Lorentz Force Law

Spring 1999

Mercer University Physics Department

Objective

In this set of experiments, a property of magnetic forces will be explored.  Among the various systems to which this concept is applicable, one of the most important is that of the electric motor.  The torque provided through the armature of a common motor derives from the magnetic part of the Lorentz force.

Equipment List

The MUL Apparatus, a power supply, a set of small ''horse-shoe'' magnets, a symmetric differential capacitive control unit, an oscilloscope, a multimeter, a 5W Power Resistor, and a current reversing switch .

Theoretical Background

The force on a charged particle moving in a magnetic field is given by:
FB = q v^B.
 		 (1)
In this equation, FB is the magnitude of the force on the charge, due to the interaction with the magnetic field, q is the charge of the particle, v^ is the component of the velocity of the particle perpendicular to the magnetic field, and B is the magnitude of the magnetic field. If there is a continuous flow of charges, this flow of charges constitutes an electric current. The force on this current, due to the interaction with the magnetic field, is:
FB = I l^ B.
 		 (2)
In this equation, I is the magnitude of the flowing current, and l^ is the component of the current perpendicular to the magnetic field.  In the present experiment, assume that this is the length of the iron pole pieces of the magnet.

In this series of experiments, a portion of a current carrying wire is immersed in a magnetic field. This wire is part of a balance which is supported by a set of posts. The magnetic force acts a distance swire from these posts, so that the force exerts a torque on the balance. A hanging weight is placed on the wire between the posts and the magnets responsible for the magnetic field. Since this hanging weight also acts a distance from the axis of rotation of the balance, it also exerts a torque on the wire. Figure 1 illustrates this geometry.

lab6-fig1b.jpg

For this set of experiments, these two torques act in opposite directions and cancel to leave the wire in static equilibrium. This static equilibrium condition allows us to write the following equation, using ds for the change in position SHMof the hanging mass, and the definition of torque t = F d^:

tclockwise = tcounter-clockwise Þ m g ds = FB swire.
 		 (3)
In this equation, m is the mass of the hanging weight, g is the acceleration of gravity, 9.8 m/s2, ds is the displacement of the hanging mass from the I = 0 equilibrium point, FB is the magnitude of the magnetic force, and swire is the distance from the axis of rotation of the balance (post indentation points) to the part of the wire in the magnetic field. Substituting into this equation the expression given for the magnetic force in equation 2,
mgds = I l^ B swire.
 		 (4)
In this set of experiments, you will test this relationship, and use it to calculate the magnetic field of the set of horse-shoe magnets.

Procedure

Dependence on Current

  1. Because of the complexity of the experimental setup, all of the equipment should already be connected. If you are having trouble with your setup, ask the lab instructor to check the setup.
  2. Using the scales provided, weigh the small piece of brass wire that is being used as the hanging weight. Record this weight on your data sheet.
  3. After weighing the hanging mass, place it on the middle notch of the MUL apparatus. This middle notch should result in an equilibrium for which the balance end-wire is centered between the pole pieces of the magnet.  If this is not the case, have your lab instructor adjust the apparatus.
  4. Measure and record the length of the wire that is immersed in the magnetic field, l^ (the length of the iron pole pieces). Also, measure the distance from the pivot (balance post indentation points) to the position of the end-wire that is immersed in the magnetic field, swire.
  5. Be sure that the ammeter is set to the 10A current position. This instrument measures the current in the wire, reading in amperes.
  6. The oscilloscope will be used to monitor the voltage from the MUL, and appropriate settings for its switches should already be established.  A voltmeter could also be employed; however, the oscilloscope is easier to use unless one has an analog as opposed to a digital voltmeter.
  7. You will first need to calibrate the MUL Force balance. Use the large knob on top of the symmetric differential capacitive control  (SDC) unit as a coarse adjustor, and the small knob on the side of the SDC unit for fine tuning.   With these controls, insure that the horizontal line on the oscilloscope is zeroed (middle of the screen).  Having done this, be careful not to jar the MUL apparatus, since jarring it will knock it out of calibration.
  8. Make sure the voltage and current controls are fully counterclockwise before turning on the power supply. With the voltage and current turned down, and with the MUL calibrated, turn on the power supply.
  9. Turn the current contol up a little (clockwise), then increase the voltage. Notice how the portion of the wire in the magnetic field moves either up or down. Notice also that the voltage trace on the oscilloscope responds to the deflection of the wire. Turn the voltage down, then flip the current reversal switch. Now increase the voltage, and note that the wire now moves in the opposite direction. This is because the current is now moving in the opposite direction to what it was before you flipped the switch. The change in direction of the current (now negative if the previous reading was positive) causes a sign change in equation 2, which in turn changes the sign on the force, and hence the type (attractive or repulsive = plus or minus) of force on the wire. Be careful not to touch the power resistor attached to the power supply; During the course of the experiment, it can get hot.
  10. With the current reversal switch in its present position, note which kind of force is acting on the wire. If the force is down, move the hanging weight one notch away from the end of the wire (i.e. away from the portion of the wire immersed in the magnetic field).  Make this adjustment with with the voltage zeroed.  If the force is repulsive, move the hanging weight one notch toward the end of the wire.
  11. Record the distance the hanging weight was moved, ds. Each of the notches are 0.5 cm away from each other; If the weight is moved toward the end of the wire, record the distance moved as a positive number; If the weight is moved away from the end of the wire, record the distance moved as a negative number. For example, if the hanging weight is moved one notch toward the end of the wire, ds would be +0.5 cm. If the hanging weight is moved two notches away from the end of the wire, ds would be -1.0cm.
  12. Adjust the current until the MUL voltage, shown on the oscilloscope, is re-zeroed. Record the current through the wire, as shown on the  ammeter.
  13. Move the hanging weight another notch and re-adjust the current until the MUL voltage is zeroed. Record the hanging weight distance and the current through the wire.
  14. Repeat this process until the hanging weight has been moved a total of 3 cm away from the equilibrium position. (Stop at 2.5 cm if the weight bumps against a fixed part of the apparatus.)
  15. Move the hanging weight back to the equilibrium position and check to see if the MUL is still zeroed. If not, re-zero it.
  16. Flip the current reversal switch so that the current through the wire is now flowing in the opposite direction (Note the algebraic sign of the current).
  17. Now move the hanging weight one notch in the opposite direction to that done previously. Adjust the current through the wire until the MUL is re-zeroed; Record the distance the weight has moved, ds (now negative) using the convention mentioned previously, and the current through the wire (also negative).
  18. Move the hanging weight an additional notch and adjust the current through the wire until the MUL is re-zeroed. Continue this process until the hanging weight has been moved 3 cm (if allowed) from the equilibrium position.

Data Analysis

Dependence on Current

  1. Graph the measured current as a function of ds. Draw in a line that you think comes closest to your data points and estimate the slope of this line. (If you do this graph with the spreadsheet, Excel (example); use 'insert trendline', employing the option that displays the slope and 'R square' as a measure of the goodness of fit.)  Use the slope of your 'best-fit' line to estimate the magnetic field , BAve , for your magnet as follows:
  2. Using Equation (4), show that the field is given by BAve = [mg / (l^swire)] / slope .
  3. Your lab instructor will measure the average field of your magnet using an instrument based on the Hall effect.  Compare your estimate of the field with the value which he measured with the Hall probe.

Questions

  1. A DC power line carries a current of 2 A due east a total distance of 4 m. What is the magnitude and direction of the magnetic force, due to the earth's field of 0.5 G (1G = 10-4 T), on the wire. Assume the magnetic field of the earth points due north. What fraction of the wire's weight is this force if the wire has a mass of 2 kg?
  2. If your estimate for BAve differed significantly from the value measured by your lab instructor, can you think of reasons why?  (NOTE:  Human error is never a satisfactory answer!)
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On 28 Dec 1998, 13:48.