University of Wisconsin Green Bay

A circular loop of wire has a radius of 0.025 m and a resistance of 3.0 Ω. It is placed in a 1.6 T magnetic field which is directed in through the loop as shown and then turned off uniformly over a period of 0.10 s. What is the current in the wire during the time that the magnetic field changes from 1.6 T to zero?

• In this problem, you are asked to find the current in a loop without a battery in it. There are two ways to cause current—you can directly drive it with a battery (or potential difference) or you can induce a current by changing the magnetic flux through a loop.

So this is an induction problem. Current in this loop is due to the changing magnetic flux through that loop.

• In induction problems, you need to work with the direction of any known vectors (such as field or current). So your picture should be a sketch of the actual situation with vectors labeled. All of this information (and more) is provided in the picture given in the problem, so no additional picture is needed.

Hint: Make sure the picture makes sense. Could you draw it from the text in the problem?

• Magnetic induction problems always begin with the definition of EMF:

EMF = -N ΔΦ/Δt = -N Δ(BA cosθ)/Δt (Calculus students use d rather than Δ.)

In this case, once you know induced EMF you will need to use it in ΔV = IR to find current. However, if you don’t recognize that step now it is fine. It will become apparent as you solve the problem. The most important thing is to recognize that you begin with induction.

Note that you will also need to use the right hand rule to determine the direction of current.

• Step 1:

Now that you know the EMF, continue to step 2 to find the current in the wire.

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Step 2:

EMF plays the same role in a circuit as potential difference (ΔV) does. Therefore,

ΔV = IR
EMF = IR
I = EMF/R = (0.031 V)(3.0 Ω) = 0.010 A

Continue to step 3 to find the direction of this current.

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Step 3:

1. The direction of the induced current is such that
2. its magnetic field opposes
3. the change in the original magnetic field.

3. The change in the original magnetic field is up out of the paper ( ∙ )
2. Opposing "out of the paper" is "into the paper" (X)
1. From the two-step right hand rule, a magnetic field into the paper is caused by a clockwise current.

• In this problem, you are asked to find the current in a wire that does not contain a battery. In general, if there is no source of voltage difference in a circuit, you should consider induction as the cause of current even though the word induction was not used in the problem. In this case, the magnetic field changes and so you have a changing magnetic flux through the loop. A changing magnetic flux results in an EMF, which has the same effect in a circuit as a voltage difference.