# A very small circular loop of area 5 × 10^-4 m^2, resistance 2 ohm and negligible self inductance initially coplaner and concentric ….

Q: A very small circular loop of area 5 × 10-4 m2, 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