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Electromagnets
Historical Note The year 1820 was an incredibly important year in the history of magnetism. In April the Danish physicist Hans Christian Ørsted (1777-1851) reported on an experiment he had done in 1819 in which a compass needle rotated when brought near a wire that was carrying electric current. His report reached Paris in September. Within a week André- Marie Ampère (1775-1836) had repeated Ørsted’s experiment and had extended the result, showing that two current-carrying wires could attract or repel each other, depending on the relative directions of the currents. Within a few months the French physicists Jean-Baptiste Biot (1774-1872) and Félix Savart (1791-1841) developed a mathematical formula for the field produced by a tiny length of wire that was similar to Coulomb’s law for electric charges. Before the year was out François Jean Dominique Arago (1786-1853) is said to have produced the first true electromagnet—a coil of wire into which an iron core could be placed to increase the magnetic strength. In this lab you will explore the magnetic fields produced by electric currents. In 1821 the Englishman Michael Faraday (1791-1867), with the help of Humphrey Davy and William Wallaston, developed a crude electric motor. Faraday later developed the concept of a magnetic field to describe the effect a magnet has on materials some distance away from it. In 1831 Faraday and the American Joseph Henry (1797-1878) independently discovered electromagnetic induction.
Goals By the end of this lab you should be able to answer these questions
- What magnetic field does an electric current produce?
- How can superposition be used to predict magnetic field strength?
- How can you produce a uniform magnetic field?
1. The field due to an infinitesimal length of wire
Introduction Biot and Savart found that the magnetic field due to an infinitesimally-short piece of wire carrying a current behaved somewhat like the electric field due to a charge. That is, it depended on the distance according to 1/ r^2. But, the direction of the field and the way the field depended on the relative directions of the current and displacement vector, were totally different. In modern notation
2
4 r
idl r dB
×
r r
Electromagnets – Version 3.0 – University of Michigan-Dearborn (^1)
d B
r
d l
r
r
r
Figure 1
In this equation i is the current in the tiny segment of wire of length dl. The length r is the distance from the current to the point where the field is being measured. The vector r ˆ^ is a unit vector that points from the current to the place that the magnetic field dB is measured. Finally, μ0/4π is a constant. The direction of the field can be found by using a standard right- hand-rule for cross-products. As with any cross product, the vector given by the cross-product, the magnetic field in this case, will be perpendicular to the plane in which the dl and r vectors rest.
Procedure Explore how the magnetic field from a short length of wire depends on direction and distance. y There is a folder called Magnetism on the computer’s desktop that contains a number of programs to be used in this lab. Open it and run the program Biot. y First, choose the direction of the wire by clicking on either the blue or green button at the top of the screen. You will see a wire made-up of six short segments. One segment, labeled dl , is colored blue to indicate that there is current in that segment. y Put the cursor on the end of the red r arrow, press the left mouse button, and drag the point. Watch how the length of the green dB vector changes and note its direction. If you are now looking at the wire in the plane, choose the other button and see the field from a wire perpendicular to the plane. (Hints on using the program—click right button and drag mouse to change viewpoint, click both buttons and drag mouse to zoom in and out.) Summarize your work:
- In what direction or directions do you point r with respect to dl to have a zero magnetic field?
- What’s the relative angle between dl and r that gives the largest magnetic field?
- Does the right-hand rule work? Check.
- How does the strength of the magnetic field depend on the distance from the current? (Hint: record the value of r and dB at one distance, then double the size of r and find the ratio of the new to the old magnetic field. Be sure that you don’t change the angle between dl and r .)
consider them separately.) Both charged objects produce an electric field everywhere, but we will be considering the field only at a single point, marked P in each case. The points P are the same distance, r , from the points marked x on each object. You should consider the ring as the rod, cut to the appropriate length, held fixed at point x , and wrapped into a circle about point P. This is useful because we want to consider how the field at point P changes as the rod is wrapped into a circle as just described.
P
x (^) x
P
r r
P
x (^) x
P
r r
a (^) b c (^) d e
f g
h
Figure 2: A charged rod and ring In each case the electric field at P can be thought of as the sum of contributions from each segment of the corresponding object. In the diagrams four representative segments of equal length on the rod and ring have been highlighted in gray and are labeled a-h. For each object, illustrate the contribution of each highlighted segment to the net field at P. To do so, if the field from a segment happens to be in the plane of the page draw an arrow with its tail attached to point P ; if the field happens to be directed into or out of the page use the symbols 9 or ☼, respectively. You should attempt to depict both the relative magnitude and direction of the field in each case. Symbolically, label these fields Ea , Eb ,... Eh.
r r r
In the same way, depict the net field at point P for each case. Keep in mind that the net field is the superposition of contributions from all segments, not just the four representative ones. Symbolically, label these fields Enet rod ,
r and Enet ring ,.
r
Check your thinking Your additions to the diagram should unambiguously illustrate the direction of each field vector. Whether you were successful in depicting their relative magnitudes or not, make your intention clear by ranking all ten vectors from greatest to least in magnitude. If any have the same magnitude, give them the same ranking. Write down your ranking in the box below.
Summarize 1. Do you predict the net electric field at point P for the ring to be greater, less or equal in magnitude to the field at point P for the rod?
- In words, describe what you think is happening to the net field at P when the rod is bent into a circle.
Prediction Two neutral conductors are shown in Figure 3 below. (The objects are not interacting with each other; consider them separately.) One is a long thin straight rod in which there is a steady current. The second is a thin ring in which there is a steady current of the same magnitude. As a result of the current, both objects produce a magnetic field everywhere, but we will be considering the field only at a single point, marked P in each case. The points P are the same distance, r , from the points marked x on each object. You should consider the ring as the rod, cut to the appropriate length, held fixed at point x , and wrapped into a circle about point P. This is useful because we want to consider how the field at point P changes as the rod is wrapped into a circle as just described.
P
x (^) x
P
r r
a (^) b c (^) d e
f g
h
i (^) i
P
x (^) x
P
r r
a (^) b c (^) d e
f g
h
i (^) i
Figure 3: There is a steady current in each object In each case the magnetic field at P can be thought of as the sum of contributions from each segment of the corresponding object. In the diagrams, four representative segments, of equal length, on the rod and ring, have been highlighted in gray and are labeled a-h. For each object, illustrate the contribution of each highlighted segment to the total field at P. To do so, if the field from a segment happens to be in the plane of the page draw an arrow with its tail attached to point P ; if the field happens to be directed into or out of the page use the symbols 9 or ☼, respectively. You should attempt to depict both the relative magnitude and direction of the field in each case. Symbolically, label these fields Ba , Bb ,... Bh.
r r r
In the same way, depict the net field at point P for each case. Keep in mind that the net field is the superposition of contributions from all segments, not just the four representative ones. Symbolically, label these fields Bnet rod ,
r and Bnet ring ,.
r
y Continue clicking to see the field at other locations in the plane of the loop other than the center of the circle. Observe at least four more locations, including one outside the loop. (Program Hint: Click on both mouse buttons and drag to zoom out so you can see the length of all vectors.) In these examples, because different loop segments are different distances from the location, the magnetic fields due to these segments vary in size, but note how in all cases the magnetic field is parallel to the central axis of the loop. y Close B_Loop_1 and start the program B_Loop_2 to examine the field on the central axis of the loop, but off the loop’s plane. y Begin clicking the left mouse button to view the field being constructed at a point on the central axis. Note that the field due to each segment is the same size, but in a different direction. Still, because of the symmetry, they add to form a field along the axis. y Continue clicking to view the field being constructed at one more location.
Summarize 1. Check the right-hand rule for individual current segments. Point the thumb of your right hand in the direction of the current of the segment. The curl of your fingers gives the sense which the field circles around the segment. Is this rule consistent with the field vectors generated by the program for points both inside and outside the loop?
- Is the field the same strength everywhere inside the loop? If not, where it is largest? Smallest?
5. Magnetic field of a helical turn
Unfortunately, we can’t measure the field in the plane of a wire loop because of the limitations of our magnetic field sensors. Instead, we’ll use a wire wound in a helix, using the Slinky®^ toy. Start the program B_Slinky_1 to explore how the field from a single helical turn differs from a loop. Click the left button to step through the calculations. Notice that the field isn’t quite along the axis and that when measuring the field near the wire the fields from individual segments point in wildly different directions. Nevertheless, the general pattern is close to that of the simple loop.
6. Measuring the field due to current in a long straight rod
Introduction We will begin the measurement part of this lab by first measuring the field produced by current in a long straight rod. As you make the measurement, keep in mind what you have learned in the preceding activities about the direction and magnitude of such a field. At the same time, anticipate the upcoming activities when the field from a loop will be measured. How will the results change?
Preparation y The magnetic field probe should be attached to a support rod as shown in the picture below. Check that the white dot is facing horizontally. (In other words, a line passing perpendicularly through the face of the white dot should be horizontal.)
Use a ruler to check the height of the center of the white dot. It should be 11.5 cm e
y rough the slot in the PVC pipe so it is positioned at
r
ongitudinal axis of the Slinky®^ pass . y
is
y connect it to the power supply.
y than a constant
l display to read Amps using the black switch to the right of the
y all four control knobs fully counterclockwise and then turn on the
y all the way clockwise. Adjust the coarse
Figure 4: Apparatus for field measurement y above the table top. Adjust it if necessary. (This height will position the probe at th vertical center of the Slinky®.) Slide the magnetic field probe th the horizontal center of the pipe. A good method is to first position the white dot so that a winding of the Slinky®^ passes right over it. Then, since the Slinky ®^ has a radius of 3.7 cm, use the ruler on the blue track to slide the sensor exactly 3.7 cm into the center. At the same time, make sure the blue support track is perpendicula to the longitudinal axis of the PVC pipe. The goal in all these steps is to have the l perpendicularly through the center of the white dot on the probe. Make it so Check that the thick copper wire is parallel with the table top and resting across the top of the PVC pipe as shown in the picture. (The goal here is to have the copper wire directly above and parallel to the plastic shaft of the magnetic field probe. Th will cause the field produced by current in the wire to be perpendicular to the face of the probe, ensuring that the total field is measured.) With an alligator clip at each end of the copper wire, Note which end is positive so you can tell the direction of current. Prepare the power supply to act as a constant current source (rather voltage source.) y Set the digita display. Start with supply. It should read 0.00 A. Turn the fine voltage control knob voltage control knob to the 9 o’clock position. The current should still read 0.00 A.
Conclusion Using your results from this and the last section calculate the ratio Bloop Brod and record
this result in the Experiment column in Table 1.
Experiment Theory
B loop Brod Table 1 What is the theoretical expectation for this ratio? Determine it using the expressions for the B -field due to current in a long rod and at the center of a current loop. Show your work in the box and record this value in the Theory column of Table 1. (You might need to consult your textbook or friendly instructor.)
Calculate the percent error between your experimental value and the theoretical value.
Was your prediction recorded in question #3 from section #3 correct? Comment.
8. From one loop to a solenoid
Introduction In this section you will examine how the magnetic field produced by a steady current in a single loop of wire changes when more loops are added.
Prediction In the previous section, as an approximation to a loop, you measured the magnetic field at the center of a single turn of a helix due to a steady current flowing in it. Consider how you would expect the B -field to change as the number of loops increases while maintaining the same current. Let’s agree to increase the number of loops symmetrically by adding the same number of loops to both the left and right of the original single loop. For instance, the first case would be to consider what happens when one loop is added to the left and the right. (Practically, this will be achieved by simply moving the two alligator clips, one turn to the left and one turn to the right, respectively.) This would create a three loop system with the magnetic field sensor at the very center. The next case would be a five loop system, created by adding another loop to the left and right. Before making the measurement, make a prediction. Circle the one graph from the selection below that you think accurately depicts the pattern that would develop if you plotted magnetic field versus number of loops.
Loops
B
-field
Loops
B
-field
Loops
B
-field
Loops
B
-field
Loops
B
-field
Loops
B
-field
Figure 6: Possible B -field vs. loops results
Check your thinking 1. Which of the following is consistent with your thinking and your choice? If none of these, add your own. (If you haven’t already, carefully consider the direction of current in each loop and the direction of the resulting B -field contributions from each loop.) A. I predict that the added loops will tend to increase the magnetic field over and above its original single-loop value. B. I predict that the added loops, because they are on both sides, will tend to cancel the original single loop magnetic field. C. I predict that the magnetic field will remain constant because the current has not changed. D. Other…
- Answer the following question in a way that is consistent with your choices above. By tripling the number of loops, for the same current, the magnetic field will A. triple B. more than triple C. increase, but less than triple D. remain constant because the current has not changed E. drop to zero because of complete cancellation F. be reduced, but the cancellation won’t be complete
- In words, provide a reasonable explanation for your prediction.
Experiment Conduct an experiment to test your prediction. Make sure the current remains constant, that you do not move the magnetic field sensor from its central position, and that you do not significantly disturb the slinky as you reposition the alligator clips. Zero the sensor when appropriate and use the averaging technique for any field measurements. Record your data in the table below and plot them using Graphical Analysis. (Print and attach the graph.)
a. The central longitudinal field
Procedure y Record the field you measured for the entire solenoid from the last row in Table 2.
B solenoid experiment = __________ μT y Determine the winding density of your solenoid by measuring its length and counting the number of loops. Use these numbers to compute n. n = ( _________ turns) ( __________m) =___________turns/m
y Compute the field of the solenoid that is given by Ampere’s law B solenoid theory = ____________ μT.
Conclusion Compare your measured result with this theoretical value by calculating a percent difference.
In the following sections you will check some of the other predictions of the solenoid
equation: B = μ 0 ni.
b. Dependence of field on current
Procedure y Measure the central longitudinal field as a function of current. Have all windings of the solenoid in the circuit. Record your results in the Table 3.
Current (A)
Magnetic Field (μT)
+3.
Table 3
Conclusion Does the field depend linearly on the current? Explain.
c. Dependence of field on winding density
Procedure y Explore qualitatively how the central longitudinal field depends on winding density. Return the current to 3.00 A. Begin collecting data with Logger Pro. While it is collecting, slowly stretch the slinky and observe the change, if any, in the field strength.
Conclusion Describe what you observed and if it is consistent with the solenoid equation, B = μ 0 ni.
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d. The off-axis interior field
Introduction The solenoid equation, B = μ 0 ni , is valid on the longitudinal axis, but not only on the
longitudinal axis. For an ideal infinitely-long solenoid it gives the correct field everywhere in the solenoid’s interior. In other words, inside an ideal solenoid, the magnetic field is uniform! Is the field inside your solenoid uniform?
Procedure y Check that all windings of the solenoid are in the circuit and the current is set at 3.00 A. y Starting on the longitudinal axis, record the field as you pull the magnetic field probe off-axis, one centimeter at a time. Zero the sensor, with the current off, before each measurement! As usual, use the averaging technique.
Distance from Central Axis (cm)
Magnetic Field (μT)
Table 4
An application What are the uses for the uniform magnetic field inside a solenoid? One important use is in MRI, magnetic resonance imaging. Solenoids used in MRI machines typically produce fields of 1.5 T or 3.0 T. The fields must be extremely uniform, differing by less than a microtesla, over the region of the body to be imaged. (This is a variation of less than 0.0001%.) The wire used is superconducting so that it has no resistance. Therefore there is no voltage across the solenoid, and no power dissipated by it.
