Question

The rotating loop in an AC generator is a square 13.0 cm on a side. It is rotated at 75.0 Hz in a uniform field of 0.800 T. Calculate the following quantities as functions of time t, where t is in seconds.

(a) the flux through the loop

mT·m²

(b) the emf induced in the loop

V

(c) the current induced in the loop for a loop resistance of 2.00 Ω

A

(d) the power delivered to the loop

W

(e) the torque that must be exerted to rotate the loop

mN·m

Answer 1

(a) We know that the flux is given by
= B.A Cos(2ft)
where f is frequency = 75 Hz
where B is magnetic field = 0.8 T
A is area = a² = (0.13)² = 0.0169 m²
= 0.8*0.0169Cos(2*75*t) = 13.52*10^-3Cos(471.24t) T-m²
= 13.52 Cos(471.24*t) mT-m²
(b) We know that
emf induced = rate of change of magnetic flux
e = -d/dt = -(d/dt)[13.52*10^-3Cos(471.24t)]
= 13.52*10^-3 *(471.24)Sin(471.24t) = 6.371Sin(471.24t) V
e =  6.371Sin(471.24t) V   
(c) We know that
I = e/R = 6.371Sin(471.24t) /2 = 3.186*Sin(471.24*t) A
(d) Power delivered
P = I²R
= [ 3.186*Sin(471.24*t)]²*2= 20.301*Sin²(471.24*t) W
(e) we know that the torque is given by
T = ABI
= (0.13)²*(0.8)*[3.186*Sin(471.24*]
= 0.0431*Sin(471.24*t) Nm
= 43.07 Sin(471.24*t) mN-m