Question

#16

A 22.0-cm-diameter coil consists of 28 turns of circular copper wire 2.6 mm in diameter. A uniform magnetic field, perpendicular to the plane of the coil, changes at a rate of 8.65 Times 10^-3 T/s. Determine the current in the loop, and the rate at which thermal energy is produced. A power line carrying a sinusoidally varying current with frequency f = 60 Hz and peak value I_0 = 55 kA runs at a height of 7.0 m across a farmer's land (Fig. 29-41). The farmer constructs a vertically oriented 2.0-m-high 10-turn rectangular wire coil below the power line. The farmer hopes to use the induced voltage in this coil to power 120-Volt electrical equipment, which requires a sinusoidally varying voltage with frequency f = 60 Hz and peak value V_0 = 170 V. What should the length l of the coil be? Would this be unethical?

Answer 1

Using Ampere's Law, we have,

2 pi.y.B = $\mu$_o I_o sinwt

=> Magnetic field B = ($\mu$_o I_o/ 2pi.y) sinwt

where y is the perpendicular distance from the power line to any point.

Therefore, the magnetic flux collected by the wire loop is:

$\Phi$ = ($\mu$_o.L / 2pi) ln(y2/y1) I_o sin(wt)

where y1 = 5 and y2 = 7 as per given.

Therefore, using Faraday's Law,

Voltage, V = ($\mu$_oL / 2pi) ln(y2/y1) w I_o cos(wt)

Solving for L we get

L = ( 2pi.Vo /$\mu$_oln(y2/y1)w.I_o)
=> L = 2pi*170/4pi*10^-7ln(7/5).2pi*60*55*10³ = 59.26m