A single circular loop of wire with radius 0020 m carries a

A single circular loop of wire with radius 0.020 m carries a current of 8.0 A is placed at the center of a solenoid that has length 0.65 m, radius 0.080 m, and 1400 turns. Determine the value of the current in the solenoid so that the magnetic field at the center of the loop is zero tesla.

Solution

magnetic field at the center of the loop due to current in the loop B = u_oi /2a

Where u _o = permeability of free space = 4(pi) x10 ^-7 H / m

           a = radius of the loop = 0.02 m

           i = Current = 8 A

Substitute values you get ,B = [4(pi)x10 ^-7 x8]/[2 x0.02]

                                          = 2.513x10 ^-4 T

If this magnetic field is equal and opposite to the magnetic field produced at the center due to solenoid then the net magnetic field at the center of the loop is zero.

So, magnetic field at the center due to solenoid B \' = B

                     u _o n i \' = B

                            i \' = B /(u _o n) where n = N /L

N = NUmber of truns = 1400

L = length = 0.65 m

So, n= N / L = 2154

Substitute values you get , i \' =(2.513x10 ^-4 )/[4(pi)x10 ^-7 x2154]

                                           = 0.0928 A