An infinite straight wire carries a current I that varies with time as shown abo

Question

An infinite straight wire carries a current I that varies with time as shown above. It increases from 0 at t = 0 to a maximum value I1 = 3.9 A at t = t1 = 15 s, remains constant at this value until t = t2 when it decreases linearly to a value I4 = -3.9 A at t = t4 = 31 s, passing through zero at t = t3 = 25.5 s. A conducting loop with sides W = 31 cm and L = 62 cm is fixed in the x-y plane at a distance d = 47 cm from the wire as shown.

1)

What is the magnitude of the magnetic flux through the loop at time t = t1 = 15 s?

2)

What is 1, the induced emf in the loop at time t = 7.5 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise.

3)

What is 2, the induced emf in the loop at time t = 17 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise.

5)

What is 4, the induced emf in the loop at time t = 28.25 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise.

Explanation / Answer

1) magnetic flux through the loop = mue*I*w/(2*pi)*ln((L+d)/d)

at t = 15s,
magnetic flux through the loop = 4*pi*10^-7*3.9*0.31/(2*pi)*ln((0.62 + 0.47)/0.47)

= 2.03*10^-7 Weber or T.m^2

2) induced emf = rate of change of magnetic flux

= mue*(dI/dt)*w/(2*pi)*ln((L+d)/d)

= 4*pi*10^-7*(3.9/15)*0.31/(2*pi)*ln((0.62 + 0.47)/0.47)

= 1.36*10^-8 V

here induced current is counter clockwise

so, emf = -1.36*10^-8 V

3) at t2 = 17 s,

induced emf = 0 (since there is no change magnetic flux through theloop)

4)

induced emf = rate of change of magnetic flux

= mue*(dI/dt)*w/(2*pi)*ln((L+d)/d)

= 4*pi*10^-7*(3.9/(25.5-15))*0.31/(2*pi)*ln((0.62 + 0.47)/0.47)

= 1.94*10^-8 V

here induced current is clockwise

so, emf = +1.36*10^-8 V

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