Question

A rectangle measuring 30.0 cm by 40.0 cm is located inside a region of a spatially uniform magnetic field of 1.25 T, with the field perpendicular to the plane of the coil (Fig.). The coil is pulled out at a steady rate of 2.00 cm/s traveling perpendicular to the field lines. The region of the field ends abruptly as shown. Find the emf induced in this coil when it is

(a) all inside the field.

(b) partly inside the field.

(c) all outside the field.

Short answer

1. The emf induced in this coil when it is all inside the field is 0(zero).
2. The emf induced in this coil when it is partly inside the field is 0.0100V.
3. The emf induced in this coil when it is all outside the field is 0(zero).

Step-by-step solution

Concept  and solution

We have a rectangle measuring 30.0 cm by 40.0 cm located inside a region of a uniform magnetic field of B = 1.25 T. The coil is pulled out at a constant speed of v = as shown in the following figure, where the magnetic field region ends abruptly.

a. First, we need to find the induced emf when the coil is all inside the field. As long as the coil is totally inside the field, the magnetic flux is constant, and since the induced emf is directly proportional to the change in the magnetic flux, then the induced emf is zero, that is,

$\epsilon =0$

b. Now we need to find the induced emf when the coil is partly inside the field. Let x be the length inside the field (along the shorter side), then the magnetic flux through the coil is given by,

role="math" localid="1664179667839" ${\varphi }_B=BA=B(0.400m)x$

thus, the induced emf is,

$$\begin{gathered} \left|\epsilon \left|=\frac{d{\varphi }_B}{dt}\right|\right|=B\frac{d(0.400m)x}{dt} \\ =B(0.400m)\frac{dx}{dt}=B(0.400m)v \end{gathered}$$

Substitute with the given we get,

$\epsilon =(1.25T)(0.400m)(0.0200m/s)=0.0100V$

c. Finally, we need to find the induced emf when the coil is totally outside the field. As long as the coil is totally outside the field, the magnetic flux is zero, and since the induced emf is directly proportional to the change in the magnetic flux, then the induced emf is zero, that is,

$\epsilon =0$

Diagram