Q: A very small circular loop of area 5 × 10^{-4} m^{2}, resistance 2 ohm and negligible self inductance initially coplaner and concentric with a much larger fixed circular loop of radius 0.1 m. A constant current of 1.0 A is passed through the bigger loop. The smaller loop is rotated with constant angular velocity ω rad/sec about it’s diameter.

Calculate the :

(a) induced emf and

(b) the induced current through the smaller loop as a function of time.

Solution : The magnetic field at the centre of the smaller loop is

$\displaystyle B = \frac{\mu_0}{4\pi}\frac{2\pi I}{a}$

B = 10^{-7} × 2 × 3.14 × 1 /0.1 = 6.28 × 10^{-6} T

Where I is the current through the larger top and a is the radius of larger loop.

The flux linked through the smaller loop at any instant is

φ = BA cos ωt

Where A is the area of the smaller loop.

The induced e.m.f. at any instant is given by

$\displaystyle e = -\frac{d\phi}{dt} $

$\displaystyle e = -\frac{d}{dt}(B A cos ωt) $

e = B A ω sin ωt

e = 6.28 × 10^{-6} × 5 × 10^{-4} ω sin ωt

e = 3.14 × 10^{-9} ω sin ωt

Current = e/R = e/2

= 1.57 × 10^{-6} ω sin ωt