Question

A small, conducting circular loop is placed inside a long solenoid carrying a current. The plane of the loop contains the axis of the solenoid. If the current in the solenoid is varied, the current induced in the loop is
(a) clockwise
(b) anticlockwise
(c) zero
(d) clockwise or anticlockwise depending on whether the resistance is increased or decreased.

Answer 1

(c) zero

The magnetic field inside the solenoid is parallel to its axis. If the plane of the loop contains the axis of the solenoid, then the angle between the area vector of the circular loop and the magnetic field is zero. Thus, the flux through the circular loop is given by
$\varphi =BA\cos \theta =BA\cos 0°=BA$
Here,
B = Magnetic field due to the solenoid
A = Area of the circular loop
θ = Angle between the magnetic field and the area vector

Now, the induced emf is given by
$$\begin{gathered} e=-\frac{d\varphi }{dt} \\ \because \phi =BA=\mathrm{constant} \\ \therefore e=0 \end{gathered}$$

We can see that the induced emf does not depend on the varying current through the solenoid and is zero for constant flux through the loop. Because there is no induced emf, no current is induced in the loop.