Induced EMF and Current in a Shrinking Loop 8 of 12 > Part A Shrinking Loop. A c
ID: 1585275 • Letter: I
Question
Induced EMF and Current in a Shrinking Loop 8 of 12 > Part A Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 165 cm, but its circumference is decreasing at a constant rate of 10.0 cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 1.00 T, which is oriented perpendicular to the plane of the loop. Assume that you are facing the loop and that the magnetic field points into the loop Find the magnitude of the emf E induced in the loop after exactly time 4.00 s has passed since the circumference of the loop started to decrease Express your answer numerically in volts to three significant figures. View Available Hint(s) Submit Part B Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field. counterclockwise Submit equest AnswerExplanation / Answer
Solution:
A)
The EMF is the time derivative of the ux. The ux is
= B.dA = BA = r^2.B
= |E| = |d/dt| = .B.2r|dr/dt| = BC(t) |dr/dt|
= Note that the information given gives us an equation for the circumference
of the circle C and the derivative of r:
C(t) = Co + dC/dt *t = (1.65m)-(0.10)t
and
dr/dt = 1/2 * dC/dt = - (0.10/2)*t
Plug this into the equation for the EMF, and the answer is
E = (t=4.0) = B(Co + dC/dt *t)|dr/dt|
= (1.00T)(1.65m-0.10m/s(9.0s))(0.10/2*) = 1.989*10^-3 m^2
B)
For the direction, the flux is decreasing into the loop (remember we're looking along the B~ field). Therefore the induced current must try to boost the flux and hence B~ ind is parallel to the field. By the RHR, that means the current is clockwise.