,OBJECT
To study the magnetic field created by a solenoid depending on the position inside a solenoid and the current through the solenoid.
APPARATUS
Solenoid, ruler, DC power supply, magnetic field sensor (Hall device), current probe, and computer.
THEORY
Electrical current flowing through a wire creates a magnetic field around the wire (see Tipler’s PHYSICS, chapter 25). If we have a circular current loop, the magnetic field in the center of the loop is
,where I is the current in the loop and R is the radius of the loop (Tipler, p. 817, eq. 25-6). Projection of the magnetic field on the axis of a current loop, parallel to the axis is given by a more complex expression:
,where x is the distance between the center of the loop and the point of observation (Tipler, p. 817, eq. 25-6).
A tightly wound solenoid can be considered as a set of circular current loops placed side by side that carry the same current. It produces a uniform magnetic field inside it. Magnetic field lines of a solenoid are shown in Figure 1.
Figure 1
Calculation of the magnetic field on the axis of the solenoid and between its ends (pp. 819-820) gives us the expression:
where n is the number of turns per unit length of the solenoid; a, b, L, and R are shown in Figure 2. N is the total number of turns in the solenoid and is related to n by n = N/L.
Figure 2
For the magnetic field at the center of the solenoid, a = b; also the length of the solenoid, L = 2a. We can also replace n with N/L. Then, the equation becomes:
(1)For a long solenoid for which L >> R, then the magnetic field at the center can be approximated by:
(2)If the origin is at one end of the solenoid, either a or b is zero. Then the equation reduces to :
(3)Thus, the magnitude of B at a point near either end of a solenoid should be about one half that at points within the solenoid away from the ends. Figure 3 gives a plot of the magnetic field on the axis of a solenoid versus position (with the origin at the center of the solenoid). The approximation that the field is constant independent of the position along the axis of the solenoid is quite good except for very near the ends. For a long solenoid (L>>R), the approximation works even better.
Figure 3.
The magnetic field sensors used in the laboratory are Hall devices, meaning that they measure the strength of a magnetic field by the Hall effect. The Hall effect is described on pages 801-803 of Tipler’s PHYSICS. A simple diagram of a Hall device is shown in Figure 4.
Figure 4
A Hall device consists of a piece of semiconductor material which is doped to create a considerable amound of free charge carriers. The device is placed in a magnetic field, so the direction of electrical current and magnetic field are perpendicular to each other. A charged particle moving through a magnetic field experiences a force described by the Lorentz equation: F = q v x B, where F is the force, q is the charge, v is the velocity, and B is the magnetic field. The direction of the force on the charged particle can be found by using the right hand rule. The direction of current corresponds to the direction of movement of positive charges, so the electrons (negatively charged particles) will move in the opposite direction. If the magnetic field is directed perpendicular to and into the plane of the page, the electrons (charge -q) moving to the left (hence a negative velocity) will experience a force upward in the plane of the page as shown in Figure 4. Since the conducting strip was initially electrically neutral, when the electrons move towards the top of the strip, positive charge develops on the bottom of the strip. A potential difference has now developed across the strip, and by measuring this voltage, the strength of the magnetic field may be determined. Hall devices are used widely, even in the ignition of automobiles.
PROCEDURE FOR EXERCISE 1
SET-UP PROCEDURE:
1. Set up the computer as instructed in Instructions for Computerized Experiments.
2. Plug the magnetic field sensor into Din 1.
3. Set the amplification of the sensor on low (switch is on the sensor box.)
4. Double click on the Vernier folder. Double click on the Phys 231 expts folder. Double click on the B vs. x-solenoid program. A message will ask you if you would like to load the calibration saved with the program. Click Yes.
DATA COLLECTION
1. Set up the apparatus as shown in Figure 5.
NOTE: The white dot on the end of the sensor must be facing away from the ring stand. The Hall device is under this dot and it should be perpendicular to the magnetic field lines inside the solenoid. The reading of the sensor is the greatest when the white dot faces magnetic north and the Hall device is positioned exactly on the axis of the solenoid.
2. Make sure that the power supply is set on 225 mA short circuit current and that the DC voltage is on the 30 V scale. Turn on the power supply and adjust the voltage to about 15 V.
3. Make sure that the magnetic field readings taken by the computer are positive values, If they are negative, simply reverse the leads to the power supply so that current is flowing through the solenoid in the opposite direction.
4. Record the position of the center of the solenoid relative to the meterstick. This point will be your origin (or x = 0 point). To the right of the center will be positive distance values and to the left will be negative distance values. When you enter a distance value, use this distance away from the origin.
5. Click on "Start". Collect data points for the values of -15 cm to 15 cm in at least 1 cm increments. In order to collect a data point:
- Enter the distance value and click OK.
- Move the sensor to the next distance value and click Keep, etc…
7. Under the Analyze Menu (on the toolbar), click Analyze Data A.
8. Highlight the data points that represent the maximum B values (the flat top of the curve.)
9. Under the Analyze Menu, click on Fit…
10. Make sure a linear equation is selected (y=b0+b1x) and click on Try Fit
11. Click on Fit Results and record the value of b0, the y-intercept of the line and its associated error. This is the experimentally determined value of B at the center of the solenoid. Report the value as B ±DB Gauss.
12. Click OK and then Cancel Fit.
13. Measure the radius of the solenoid and the length of the solenoid and record thier values. For the radius, measure the inner and outer radii and take the average.
14. Record the value of the magnetic field at both ends of the solenoid.
15. Quit the program by clicking on the boxes in the upper left-hand corner. Don’t save anything!
DATA ANALYSIS FOR EXERCISE 1
1. Sketch your curve.
2. Calculate the value of I in your circuit using the Ohm’s law. The value for R is given on your solenoid.
3. Calculate the theoretical value of B at the center of the solenoid using equation (1) and the value obtained for I. Make sure that all values you use are in the SI system. Convert your result for the magnetic field into Tesla. (SI system unit for magnetic field.) 1 T = 104 Gauss. Does the theoretical value equal the experimental value within the experimental error?
4. Calculate the theoretical value of B at the ends of the solenoid using equation (3). How does this value compare with the experimentally determined value?
5. Calculate the value of the magnetic field of a long solenoid using
equation (2). How well does this simple solenoid equation predict the magnitude
of the magnetic field in this experiment? Would you expect the values to
agree? If so, where?
PROCEDURE FOR EXERCISE 2
1. In the Phys 231 labs folder, open the program B vs. I. Respond yes to loading the calibration saved with the experiment.
2. Take a current sensor and plug it in Probe 1 of the Dual Channel Amplifier. Plug Din 1 on the Dual Channel Amplifier into Din 2 on the ULI Box. The Magnetic Field sensor should be plugged into Din 1.
3. Set up the circuit as shown in Figure 6.
Figure 6.
4. Set the DC voltage supply on 225 mA short circuit and the 10 V scale.
5. Place the magnetic field sensor in the exact center of the solenoid.
6. Click on Start.
7. Slowly turn the voltage adjust knob between 0 and 10 V.
8. Click on Stop. Turn the Voltage Adjust Knob to 0 V.
9. On the Analyze menu (on the toolbar) select Fit…
10. Make sure a linear fit is chosen (y = b0 + b1x). Click on Try Fit. Record the equation.
11. Click on Cancel Fit
12. Repeat steps 6-9 five times.
13. Shut down the computer as instructed in Instructions for Computerized Experiments.
14. Disconnect all circuits and leave the materials as they were
found on the lab bench.
DATA ANALYSIS FOR EXERCISE 2
1. The slope of your graph of B vs. I is given by
from equation (1). Determine the value of m0 for each of your trials. Be very careful, the units of slope of the line on the graph are in Gauss/A. Convert them to T/A.
2. Find the average value and standard deviation of your experimentally determined value of m0. Report the result as m0 = m0AVG ±Dm0, where Dm0 is the standard deviation of the trials.
3. Compare your experimentally determined value with the accepted value of m0, which can be found in the back of Tipler.
Make sure to include all of the original lab data that was generated. Be sure to show your calculations for the data analysis questions which require them. In your conclusions, make sure you include the answers to the questions in the data analysis for exercise 1 (# 3, 4, and 5) and exercise 2 (# 2 and 3). Evaluate possible sources of error in the experiment.