Apacked bundle of 100 long, straight, insulated wires forms a cylinder of radius r 0.500 cm. (a) if each wire carries 2.00 a, what are the magnitude and direction of the magnetic force per unit length acting on a wire located 0.200 cm from the center of the bundle? (b) what if? would a wire on the outer edge of the bundle experi- ence a force greater or smaller than the value calculated in part (a)? give a qualitative argument for your answer.

(a) and is directed perpendicular to both the magnetic field and force.

(b) Greater

Solution:

As per the question:

No. of wires, N = 100

Radius of the cylinder formed, R = 0.500 cm = 0.005 m

Current in each wire, I = 2 A

Distance from the center, d= 0.200 cm = 0.002 m

Now,

(a) For the magnitude of the magnetic force:

We use Ampere circuital Law, to calculate the magnetic field at a distance d = 0.002 m of an imaginary cylinder:

where

= permeability in free space

B = Magnetic field

            (1)

where

where

A = Area =

J =

Thus

     (2)

Using eqn (1) and (2):

Now,

The magnetic force is given by:

For maximum force:

Thus force per unit length:

Thus

(Since, )

The force is directed perpendicular to the direction of the magnetic field and force.

(b) We know that the distance over which the force acts is direct proportion with the force:

Force, F ∝ Distance, d

Thus the outer edge of the bundle will experience a greater force than that calculated in part (a)