A power line carrying a sinusoidally varying current with frequency f=60 Hz and

Question

A power line carrying a sinusoidally varying current with frequency f=60 Hz and peak value I0 = 55kA runs at a height of 7.0m across a farmer's land. The farmer constructs a vertically oriented 2.0m 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 sinusiodally varying voltage with frequency and peak value V0 = 170V. What should the length l of the coil be? (Hint: Start with the magnetic field from the power line. Next, find the flux in the loop. This requires integration because the magnetic field is not constant through the loop. Then use Faraday's law with the flux to find the induced EMF. You can then calculate for the length.)

Explanation / Answer

Assume that the power line is very long compare to length of the coil.
So B=muy*I/(2*pi*r).
The magnetic flux.
dF=B*l*dr=muy*I*L/(2pi) *dr/r
where L be the length of the coil.
integrate we have.
F=(muy*I*L/(2pi)) *ln(r1/r2)
F=muy*I0*cos(wt)*L*ln(r1/r2)/(2pi) (where w=2pi*f)
Take the derivative of F.
F'=U=muy*I0*w*sin(wt)*L*ln(7/5)/2pi
so V0=muy*I0*w*ln(7/5)*L/2pi
V0=muy*I0*f*ln(7/5)*L
so L=V0/(muy*I0*f*ln(7/5))=122 (m).

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