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A thin dielectric disc uniformly distrib...

A thin dielectric disc uniformly distributed with charge q has radius r and is rotated n times per second about an axis perpendicular to the disc and passing through the centre. Find the magnetic induction at the centre of the disc.

Text Solution

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Consider a hypothetical ring of radius x and thickness dx of a disc as shown in the figure.

Charge on the ring.
`dq=q/(pir^2)xx(2pi"x"xxdx)`
Current due to rotation of charge on ring is
`di=(dq)/(T) = ndq`
`implies di =(nq2x dx)/(r^2)`
Magnetic field at the centre o due to current of the ring element.
`dB=(mu_0di)/(2x)= (mu_0nq2xdx)/(r^2(2x))`
`dB = (mu_0ndxq)/(r^2)`
`int_0^B dB = int_0^(r)(mu_0ndx q)/(r^2)`
`B = (mu_0nq)/r^2 int_0^r dx`
`B = (mu_0 nq)/r^2 xxr`
`B = (mu_0 nq)/(r)`
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