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 A at t = t1 = 13 s, remains constant at this value until t = t2 when it decreases linearly to a value I4= -5 A at t = t4 = 28 s, passing through zero at t = t3 = 23 s. A conducting loop with sides W = 38 cm and L = 54 cm is fixed in the x-y plane at a distance d = 25 cm from the wire as shown.

1)

What is the magnitude of the magnetic flux ? through the loop at time t = t1 = 13 s?

T-m2

2)

What is ?1, the induced emf in the loop at time t = 6.5 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 = 15 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 = 23 seconds?

Clockwise

Counterclockwise

There is no induced current at t = t3

5)

What is ?4, the induced emf in the loop at time t = 25.5 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

Explanation / Answer

Given that

A conducting loop with sides W = 38 cm=0.38mand L = 54 cm=0.54m is fixed in the x-y plane at a distance d = 25 cm=0.25m from the wire

a)

The magnitude of the magnetic flux through the loop at time t = t1 = 13 s

=integral B.dA =integral(uoi/2pix)w.dx) =uoi/2piintegral(1/x)dx

integrating from the limits d to d+L

=(uoiw/2pi)*ln((d+L)/(d))

=(2*10-7*5*0.38m)*ln(0.25+0.54/0.25) =4.372*10-7Wb

b)

What is 1, the induced emf in the loop at time t = 6.5 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise. is given by

e=-d /dt =(0.8744*10-7Wb)(5/13) =-0.336*10-7V Counter clockwise direction

c)

What is 2, the induced emf in the loop at time t = 15 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise.

e =0V

d)

What is 4, the induced emf in the loop at time t = 25.5 s? Define the emf to be positive if the induced current in the loop is clockwise and negative if the current is counter-clockwise.

e=-d /dt =4.372*10-7Wb(di/dt) =4.372*10-7Wb(5/25.5) =0.857*10-7V

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