A charge Q is uniformly distributed over the surface of… | JEE Main 2012 (AIEEE)

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Biot-Savart Law & Circular Coils

A charge $Q$ is uniformly distributed over the surface of non-conducting disc of radius $R.$ The disc rotates about an axis perpendicular to its plane and passing through its center with an angular velocity $\omega.$ As a result of this rotation a magnetic field of induction $B$ is obtained at the center of the disc. If we keep both the amount of charge placed on the disc and its angular velocity to be constant and vary the radius of the disc then the variation of the magnetic induction at the center of the disc will be represented by the figure:

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Answer: (a)

A ring of radius $x$ , width $dx$ carries $dI = \dfrac{\sigma\,2\pi x\,dx\,\omega}{2\pi} = \sigma\omega x\,dx$ and gives $dB = \dfrac{\mu_0\,dI}{2x} = \dfrac{\mu_0\sigma\omega}{2}dx$ .
$B = \dfrac{\mu_0\sigma\omega R}{2}$ with $\sigma = \dfrac{Q}{\pi R^2}$ , so $B = \dfrac{\mu_0 Q\omega}{2\pi R}$ .
With $Q$ and $\omega$ fixed, $B \propto \dfrac{1}{R}$ : a rectangular hyperbola.
So the answer is (a).

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Biot-Savart Law & Circular Coils question from JEE Main 2012 (AIEEE)

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