Question

A circular loop of flexible iron wire has an initial circumference of 164cm , but its circumference is decreasing at a constant rate of 11.0cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 1.00T , 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 induced in the loop after exactly time 4.00s has passed since the circumference of the loop started to decrease AND find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

Answer 1

emf = 0.02525 V

induced current with a counterclockwise direction

Step-by-step explanation:

The emf is given by the following formula:

`emf=-(\Delta \Phi_B)/(\Delta t)=-B(\Delta A)/(\Delta t)`
`\ \ =-B(A_2-A_1)/(t_2-t_1)` (1)

ФB: magnetic flux = BA

B: magnitude of the magnetic field = 1.00T

A2: final area of the loop; A1: initial area

t2: final time, t1: initial time

You first calculate the final A2, by taking into account that the circumference of loop decreases at 11.0cm/s.

In t = 4 s the final circumference will be:

`c_2=c_1-(11.0cm/s)t=164cm-(11.0cm/s)(4s)=120cm`

To find the areas A1 and A2 you calculate the radius:

`r_1=(164cm)/(2\pi)=26.101cm\\\\r_2=(120cm)/(2\pi)=19.098cm`

r1 = 0.261 m

r2 = 0.190 m

Then, the areas A1 and A2 are:

`A_1=\pi r_1^2=\pi (0.261m)^2=0.214m^2\\\\A_2=\pi r_2^2=\pi (0.190m)^2=0.113m^2`

Finally, the emf induced, by using the equation (1), is:

`emf=-(1.00T)((0.113m^2)-(0.214m^2))/(4s-0s)=0.0252V=25.25mV`

The induced current has counterclockwise direction, because the induced magneitc field generated by the induced current must be opposite to the constant magnetic field B.