An infinite straight wire carries a current I that varies with time as shown abo
ID: 1435155 • Letter: A
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 = 4.1 A at t = t1 = 19 s, remains constant at this value until t = t2 when it decreases linearly to a value I4 = -4.1 A at t = t4 = 37 s, passing through zero at t = t3 = 30.5 s. A conducting loop with sides W = 28 cm and L = 52 cm is fixed in the x-y plane at a distance d = 31 cm from the wire as shown.
1)
What is the magnitude of the magnetic flux ? through the loop at time t = t1 = 19 s?
T-m2
2)
What is ?1, the induced emf in the loop at time t = 9.5 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise.
V
3)
What is ?2, 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.
V
4)
What is the direction of the induced current in the loop at time t = t3 = 30.5 seconds?
Clockwise
5)
What is ?4, the induced emf in the loop at time t = 33.75 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise.
V
110 I(t) 0 -rExplanation / Answer
1) Use the Biot-Savart for an infinitely long wire.
B = (µ x I) / (2 x r)
Ultimately, we're looking to integrate the magnetic field with respect to the area of the loop. Since the magnetic field is affected by the radius, while the current at this point is a constant, we know we're going to have to integrate 1/r, which in this case is from 0.31 - 0.83 m. Width in this case does not directly affect the magnetic field. The equation becomes:
[int] B ds = ((µ x A) / (2)) x [int, A to B] 1/r dr x W = 2.26*10^-7
2) 1 =-d(phi)/dt = (µW/2pi)*(ln(d+L)/d)(dI/dt) = 1.19*10^-8 V
3) 2 = 0
4) Clockwise
5) 4 = -d(phi)/dt = (µW/2pi)*(ln(d+L)/d)(dI/dt) = 6.7*10^-9 V