SAT Physics Subject Test
Chapter 12 Electromagnetic Induction
FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION
Electromotive force can be created by the motion of a conductor through a magnetic field, but there’s another way to create an emf from a magnetic field.
The magnetic flux, ΦB, through an area A is equal to the product of A and the magnetic field perpendicular to it: ΦB = B ┴ A = BA cosθ. Magnetic flux measures the density of magnetic field lines that cross through an area. (Note that the direction of A is taken to be perpendicular to the plane of the loop.)
The figure shows two views of a circular loop of area 30 cm2 placed within a uniform magnetic field, B (magnitude 0.2 T).
1. What’s the magnetic flux through the loop?
2. What would be the magnetic flux through the loop if the loop were rotated 60°?
3. What would be the magnetic flux through the loop if the loop were rotated 90°?
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1. Since B is parallel to A, the magnetic flux is equal to BA.
The SI unit for magnetic flux, the tesla·meter2, is called a weber (abbreviated Wb). So ΦB = 6 10–4 Wb.
2. Since the angle between B and A is 60°, the magnetic flux through the loop is
ΦB = BA cos60° = (6 × 10–4 Wb)(0.5) = 3 × 10–4 Wb
3. If the angle between B and A is 90°, the magnetic flux through the loop is zero, since cos 90° = 0.
Even though ΦB is not a vector, we can think of it as having a direction, the direction A.
For the SAT Physics Subject Test, remember that changes in magnetic flux induce emf. According to Faraday’s law of electromagnetic induction, the emf induced in a circuit is equal to the rate of change of the magnetic flux through the circuit. This can be written mathematically as
This induced emf can produce a current, which will then create its own magnetic field. The direction of the induced current is determined by the polarity of the induced emf and is given by Lenz’s law (which also explains the minus sign in the equation above): The induced current will always flow in the direction that opposes the change in magnetic flux that produced it. If this were not so, then the magnetic flux created by the induced current would magnify the change that produced it, and energy would not be conserved. Note that it is common practice to refer to the “direction” of flux. Keep in mind that flux is scalar, and so has no direction. Talking about flux having a direction makes applying Lenz’s law easier, but what we are really referring to is the direction of the field producing the flux.
The circular loop of Example 14.1 rotates at a constant angular speed through 60° in 0.5 s.
4. What’s the induced emf in the loop?
5. In which direction will current be induced to flow?
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4. As we found in Questions 1–3, the magnetic flux through the loop changes when the loop rotates. Using the values we determined earlier, Faraday’s law gives
5. The original magnetic flux was 6 × 10–4 Wb upward, and was decreased to 3 × 10–4 Wb. So the change in magnetic flux is –3 × 10–4 Wb upward, or, equivalently, ∆ΦFB = 3 × 10–4 Wb, downward. To oppose this change, we would need to create some magnetic flux upward. The current would be induced in the counterclockwise direction (looking down on the loop) because the right-hand rule tells us that then the current would produce a magnetic field that would point up.
Current will flow only while the loop rotates, because emf is induced only when magnetic flux is changing. If the loop rotates 60° and then stops, the current will disappear.
6. Consider the conducting rod that’s moving with constant velocity v along a pair of parallel conducting rails (separated by a distance ℓ), within a uniform magnetic field, B.
Find the induced emf and the direction of the induced current in the rectangular circuit.
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The area of the rectangular loop is x, where x is the distance from the left-hand bar to the moving rod.
Because the area is changing, the magnetic flux through the loop is changing, which means that an emf will be induced in the loop. To calculate the induced emf, we first write ΦB = BA = Bℓx, then since ∆x/∆t = v (distance/time = speed), we get
We can figure out the direction of the induced current from Lenz’s law. As the rod slides to the right, the magnetic flux into the page increases. How do we oppose an increasing into-the-page flux? By producing out-of-the-page flux. For the induced current to generate a magnetic field that points out of the plane of the page, the current must be directed counterclockwise (according to the right-hand rule).
Notice that the magnitude of the induced emf and the direction of the current agree with the results we derived earlier, in the section on motional emf.
This example also shows how a violation of Lenz’s law would lead directly to a violation of the law of conservation of energy. The current in the sliding rod is directed upward, as we saw from using Lenz’s law, so the conduction electrons are drifting downward. The force on these drifting electrons—and thus the rod itself—is directed to the left, opposing the force that’s pulling the rod to the right. If the current were directed downward, in violation of Lenz’s law, then the magnetic force on the rod would be to the right, causing the rod to accelerate to the right with ever-increasing speed and kinetic energy, without the input of an equal amount of energy from the outside.
A permanent magnet creates a magnetic field in the surrounding space. The end of the magnet at which the field lines emerge is designated the north pole (N), and the other end is the south pole (S).
7. The figure on the next page shows a bar magnet moving down, through a circular loop of wire. What will be the direction of the induced current in the wire?
8. What will be the direction of the induced current in the wire if the magnet is moved as shown in the following diagram?
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7. The magnetic flux down through the loop increases as the magnet is moved. By Lenz’s law, the induced emf will generate a current that opposes this change. How do we oppose a change of more flux downward? By creating flux upward. So, according to the right-hand rule, the induced current must flow counterclockwise (because this current will generate an upward-pointing magnetic field). As the magnet is moved closer to the loop, the magnitude of the magnetic field moving through the loop increases, as magnetic field is inversely proportional to distance.
8. In this case, the magnetic flux through the loop is upward and, as the south pole moves closer to the loop, the magnetic field strength increases so the magnetic flux through the loop increases upward. How do we oppose a change of more flux upward? By creating flux downward. Therefore, in accordance with the right-hand rule, the induced current will flow clockwise (because this current will generate a downward-pointing magnetic field).