The linear charge density on a dielectric ring of radius R is varying with θ as λ=λ0cos⁡θ2 where λ0 is a constant. Find the potential at the centre O of the ring (in volt).

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The linear charge density on a dielectric ring of radius R is varying with $θ$ as $λ = λ_{0} \cos \frac{θ}{2}$ where $λ_{0}$ is a constant. Find the potential at the centre O of the ring (in volt).

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Detailed Solution

Potential is a scalar quantity. The charge density function suggests that net charge is zero.

Net charge on the ring

$\begin{array}{l} Q = \int_{0}^{2 π} λRdθ \\ = λ_{0} R \int_{0}^{2 π} \cos (θ / 2) dθ = 0 \\ V = \frac{1}{4 {πε}_{0}} (\frac{Q}{R}) = 0 \end{array}$

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