A circular loop of flexible iron wire has an initial circumference of 170 cm , but its circumference is decreasing at a constant rate of 12.0 cm/s due to a tangential pull on the wire. The loop is in a constant uniform magnetic field of magnitude 0.800 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.
(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.

after 9.00 s ; the magnitude of the induced emf = 0.000944 V

The induced current will be in the clockwise direction.

Explanation:

To find the circumference of the circle ; we use the expression:

C = 2πr

differentiating the above expression to determine the rate of change of the circumference; we have:

Here ; the rate of change of the circumference is

Replacing 12.0 cm/s for ; we have :

However ;the area of the circle is expressed as:

A = πr²

Also; the rate of change of the area is determined as;

Here; the rate of change of area is

Replacing - 12.0 cm/s for in equation (2) ; we have:

Now; after 9.00 s ; the value of circumference of the loop is decreased by:

= 1.08 m

The expression for the circumference of the loop after 9.00 s is:

Given that :

C = ( 1.7 - 1.08) m

C = 0.62 m

Recall that :

C = 2πr

0.62 = 2×3.14 × r

r =

r = 0.099 m

Replacing r = 0.099 m into equation (3)

From Faraday's law, Induced emf (ε) is expressed as:

and the magnetic flux is given as:

replacing the value of into above equation; we have:

=

where θ = 0 ; B = 0.800 T and = -0.0118 m²/s

Therefore; after 9.00 s ; the magnitude of the induced emf = 0.000944 V

b) The magnitude of id directing into the plane; However ; considering Lenz's law ; it states that the changes produced in the field will be opposed by the induced current. Thus ; it is found that the direction of the current will be in clockwise direction.

magnityde and direction