Question

A current with sinusoidal time-dependence, whose peak value is 8.55 A and frequency is 671 cycles per second, passes through a long straight wire. A rectangular 542 ㎝ ×24.4 cm loop of copper wire is located 60.2 cm from the wire and is coplanar with it. Calculate the average power dissipated by the loop, if its resistance is 54.3 Ω. ( 60.2 cm 24.4 cm Number 54.2 cm

A current with sinusoidal time-dependence, whose peak value is 8.55 A and frequency is 671 cycles per second, passes through a long straight wire. A rectangular 54.2 cm × 24.4 cm loop of copper wire is located 60.2 cm from the wire and is coplanar with it. Calculate the average power dissipated by the loop, if its resistance is 54.3 Ω.

Answer 1

r = 60.2 cm

w = width of loop = 54.2 cm

L = length of loop = 24.4 cm

i(t) = i_o Sin(2 $\pi$ ft) = 8.55 Sin(2 x 3.14 x 671 t) = 8.55 Sin(4213.88 t)

magnetic flux through the loop is given as

$\phi$ = ( $\mu$ /4 $\pi$ ) (2 i(t) L) log((r + w)/r)

taking derivative both side

E = induced emf = - d $\phi$ /dt =  ( $\mu$ /4 $\pi$ ) (2 i(t) L) log((r + w)/r)

E = (10^-7) (2 x 8.55 x 4213.88 x 0.244) Cos(4213.88t) log((60.2 + 54.2)/60.2)

E = 0.00113 Cos(4213.88t)

average power is given as

P = (0.00113)²/54.3 = 2.4 x 10^-8 Watt