Question

Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 167 cm , but its circumference is decreasing at a constant rate of 13.0 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.

(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

a) fem = - 0.0103 V, b) the applied field is in a vertical upward direction, the induced current is clockwise.

Step-by-step explanation:

a) For this exercise we use Faraday's law

fem =
`- (d \phi)/(dty)`

the magnetic flux is

Ф = B. A = B A cos θ

The bold letters indicate vectors, in this case the direction of the magnetic field and the normal to the circumference is parallel therefore the angle is zero and the cos 0 = 1

fem = - B dA / dt

the area of ​​a circle is

A = π r²

l

et's perform the derivative

dA / dt =π 2r
`(dr)/(dt)`

we substitute

fem = - B 2π r
`(dr)/(dt)`

the circumference of a circle is

L = 2π r

we substitute

fem = - B L ( L )

fem =

Let's find the circumference for the 9 s, let's use a direct rule of proportions

If the circumference changes 13cm at t = 1. how much does it change at t=9s

ΔL = 13cm (9s / 1s) = 117cm

the circumference that is

L = Lo - ΔL

L = 167 - 117

L = 50 cm

let's reduce all magnitudes to the SI system

L = 0.50 m

= 0.130 m / s

calculate us

fem = - 1.00 0.50 0.130

fem = - 0.0103 V

b) the electromotive force induced in the opposite direction to the change of the radius and is decreasing with time, the current follows the direction of the decreased voltage therefore the current is induced in the opposite direction to the change of the magnetic flux.

If the applied field is in a vertical upward direction, the induced current is clockwise.