Question

8-7. A large number N of closely spaced turns of fine wire are wound in a single layer upon the surface of a wooden sphere of radius a, with the planes of the turns perpendicular to the axis of the sphere and completely covering its surface. If the current in the winding is i , determine the magnetic field at the center of the sphere.

Answer 1

Solution:

No. of turns = N

Radius = a

Current in the winding = i

No. Of turns per unit length = N/2a

No of turns in each segment making an angle d = N/2a * a d = N d/2

Magnetic field at the center of the loop = dB =

(( N/2 x d ) o/2 * I a^2 cos^2 ) /(a^2 cos^2 +a^2 sin^2 )^1/3

= o N i a^2 /4 a^3 - pi /2 to pi/2  (cos^2 )  d =  o N i /8 a

For the complete circle, this will be 2 * uo N i / 8a

B = o N i / 4 a

Answer 2

Solution:

No. of turns = N

Radius = a

Current in the winding = i

No. Of turns per unit length = N/2a

No of turns in each segment making an angle d = N/2a * a d = N d/2

Magnetic field at the center of the loop = dB =

(( N/2 x d ) o/2 * I a^2 cos^2 ) /(a^2 cos^2 +a^2 sin^2 )^1/3

= o N i a^2 /4 a^3 - pi /2 to pi/2  (cos^2 )  d =  o N i /8 a

For the complete circle, this will be 2 * uo N i / 8a

B = o N i / 4 a