A conducting square loop is placed in a magnetic field B with its plane perpendicular to the field. Some how the sides of …..

Q. A conducting square loop is placed in a magnetic field B with its plane perpendicular to the field. Some how the sides of the loop start shrinking at a constant rate α  . The induced emf in the loop at an instant when its side is a, is

(a) 2a α B

(b) a² α B

(c) 2a² α B

(d) a α B

Ans:(a)

Sol: At any time t ,

Side of Square l = (l₀ – α t ) = a (given)

Area = S² = l² = (l₀ – α t)²

Magnetic Flux = BS = B (l₀ – α t)²

Induced emf in the loop , $\displaystyle e = -\frac{d\phi}{dt}$

$ \displaystyle e = -\frac{d}{dt} (B (l_0 – \alpha t)^2)$

$ \displaystyle e = – 2 B(l_0 – \alpha t)(-\alpha) $

$ \displaystyle e = 2 B a \alpha $