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 = 5.6 A at t = t1 = 10 s, remains constant at this value until t = t2 when it decreases linearly to a value I4 = -5.6 A at t = t4 = 23 s, passing through zero at t3 = 19 s. A conducting loop with sides Wcm = 24 cm and Lcm = 72 cm is fixed in the x-y plane at a distance dcm = 42 cm from the wire as shown.

t2 not defined as a variable name.          muo = 4pi * Bconst

1) What is the magnitude of the magnetic flux ? through the loop at time t = t1= 10 s?

2)What is ?1, the induced emf in the loop at time t = 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 the direction of the induced current in the loop at time t = t3= 19 seconds?

a) Clockwise

b) Counterclockwise

c) There is no induced current at t = t3

4) What is ?4, the induced emf in the loop at time t = 21 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

Let the loop dimensions be Lin the direction of the wire and w in the direction perpendicular to the wire. The loop extends from d to d+ w in the direction perpendicular to the wire - call that the y-direction.

The magnetic flux P = integral(B dA)

Now dA = dx dy and B = uo*I/y so

P = integral[d, w+d](integral(0, L) u0*I/y dx) dy) = integral[d, w+d](L*u0*I/y dy) = L*u0*I*ln((d+w)/w)

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