An infinite straight wire carries a current I that varies with time as shown abo
ID: 2183350 • 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= 2.7 A at t = t1= 10.0 s, remains constant at this value until t = t2when it decreases linearly to a value I4= -2.7 A at t = t4= 29.0 s, passing through zero at t = t3= 22.0 s. A conducting loop with sides W = 20.0 cm and L = 58.0 cm is fixed in the x-y plane at a distance d = 52.0 cm from the wire as shown.
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= 2.7 A at t = t1= 10.0 s, remains constant at this value until t = t2when it decreases linearly to a value I4= -2.7 A at t = t4= 29.0 s, passing through zero at t = t3= 22.0 s. A conducting loop with sides W = 20.0 cm and L = 58.0 cm is fixed in the x-y plane at a distance d = 52.0 cm from the wire as shown. 1) What is the magnitude of the magnetic flux ? through the loop at time t = t1= 10.0 s? 2)What is ?1, the induced emf in the loop at time t = 5.0 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 = 12.0 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) Use the Biot-Savart for an infinitely long wire. B = (µ x I) / (2p 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 .26 - .84 m. Width in this case does not directly affect the magnetic field. The equation becomes: [int] B ds = ((µ x A) / (2p)) x [int, A to B] 1/r dr x W 2) You know the total flux at the 15 seconds, and you can see that it increased linearly. Therefore, the EMF (-dF/dt), also acts linearly. Simply divide the total flux you calculated at 15 seconds by the time it took to get there, and you'll get the EMF along that entire first section. Note: Lenz's law tells us that a change in flux will generate a current that flows in such a way to oppose the change in flux. Use the RHR to determine which way the induced current will flow, thus determining your sign. 3) There is no change in magnetic flux, so this should be pretty straight forward.