Question

Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 165 cmcm , but its circumference is decreasing at a constant rate of 14.0 cm/scm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 0.800 TT , 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.
(a) Find the emf induced in the loop at the instant when 9.0 s have passed.
(b) Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

Answer 1

Answer:

(a)  emf = 1.18 mV

(b) counter-clockwise sense

Explanation:

(a) The induced emf is given by the following formula:

$emf=-\frac{d\Phi_B}{dt}$     (1)

where:

ФB: magnetic flux = AB = (area of the loop)*(magnitude of the magnetic field)

A = πr^2

B = 0.800 T

You replace the expression for the magnetic flux in the equation (1):

$emf=-B\frac{\Delta A}{\Delta t}=-B\frac{A_2-A_1}{t_2-t_1}$

A1: initial area

A2: final area

t2-t1: time interval  = 9.0s

Then you have to calculate the change in the area of the loop, by using the information about the circumference of the loop. First you calculate the radius of the loop for a circumference of 165 cm = 1.65m

$s=1.65m=2\pi r\\\\r=\frac{1.65m}{2\pi}=0.262m$

You calculate the initial area A1:

$A_1=\pi (0.262m)^2=0.215m^2$

After 9.0 second the circumference will be:

$s'=1.65m-0.14\frac{m}{s}(9.0s)=0.39m$

the new radius and the final area is:

$r=\frac{0.39m}{2\pi}=0.062m$

$A_2=\pi(0.062m)^2=0.012m^2$

Finally, you replace in the equation (1):

$emf=-(0.800T)\frac{0.012m^2-0.215m^2}{9.0s}=1.8*10^{-3}V=1.8mV$

The induced emf in the circular loop is 1.18mV

(b) The induced emf generates an electric current, which produces a magnetic field that is opposite to the direction of the constant magnetic field of 0.800T. Due to this magnetic field point into the loop. The current has to have a direction in a counter-clockwise sense.