A thin wire is wound very tightly in one layer on the surface of a sphere of paramagnetic material `(mu_(r)~~1)`. The planes of all the turns can be assumed to be perpendicular to the same diameter of the sphere. The turns over the entire surface of the sphere. The radius of the sphere is `R`, the total number of turns is `N`, and the current in the winding is `I`. Find the magnetic induction at the centre of the sphere. (Given `mu_(0)NI=20R`)

Explanation

Consider a circular ring of radius `'a'` at a distance `x` from the centre `O` such that
`a=Rsintheta`
`x=Rcostheta`
The number of turns `dN` in the length `Rd theta` is
`dN=(N)/(piR)(R d theta)=(N)/(pi)d theta`
Magnetic induction at the centre of the sphere due to the ring is `dB=(mu_(0)N)/(pi)(d thetaIx^(2))/(2(a^(2)+x^(2))^(3//2))dx=(mu_(0))/(2pi)(NI)/(R )sin^(2)thetad theta`
On integrating
`B=(mu_(0)NI)/(2piR)int_(0)^(pi)sin^(2)theta d theta`
`B=(mu_(0)NI)/(2piR)[(pi)/(2)]` or `B=(mu_(0)NI)/(4R)`