A loop of wire is at the edge of a region of space containing a uniform magnetic field B⃗ . The plane of the loop is perpendicular to the magnetic field. Now the loop is pulled out of this region in such a way that the area A of the coil inside the magnetic field region is decreasing at the constant rate c. That is, dAdt=−c, with c>0.

QUESTION:

Part A

The induced emf in the loop is measured to be . What is the magnitude of the magnetic field that the loop was in?

Part B

For the case of a square loop of side length being pulled out of the magnetic field with constant speed (see the figure), what is the rate of change of area ?

Part A:

Part B:

Explanation:

Part A:

Faraday's law says that the induced voltage is equal to

,

which in our case(because we have only one loop) becomes

,

and since the magnetic field is uniform (not changing),

Now, we know that

therefore,

which gives us

Part B:

The area of the loop can be written as

,

where is the instantaneous length of the side along which the loop is moving.

Taking the derivative of both sides we get:

,

and since we have

where the negative sign indicates that the area is decreasing.