A charge Q is uniformly spread over a disc of radius R made from nonconducting material. This disc is rotated about its geometrical axis with frequency f. Find the magnetic field produced at the centre of the disc.

Explanation

1. Suppose the disc with radius R is divided into the concentric rings with various radii. Consider one of these rings with radius r and width dr. Total charge on the disc is Q.
Hence charge per unit area = `Q/(piR^(2))`
`therefore` The charge on the ring with radius
r = (area of the ring) (charge per unit area)
= `(2pir*dr)(Q/(piR^(2)))`
2. If the ring is rotating with frequency f, they current produced `I=Q/(piR^(2))2pir*dr*f` any magnetic field produced at the centre due to this current `dB=(mu_(0)I)/(2r)=mu_(0)Q/(piR^(2))2pir(dr)/(2r)`
`f=(mu_(0)Qf)/R^(2)*dr`
`therefore` Magnetic field B produced at the centre due to the whole disc
`B=intdB=int_(0)^(R)(mu_(0)Qf)/R^(2)dr=(mu_(0)Qf)/R^(2)int_(0)^(R)dr`
`thereforeB=(mu_(0)Qf)/R`