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Total Charge of a Circular Wire

Try this problem, useful for Physics Olympiad based on Total Charge of a Circular Wire.

The Problem:

A circular wire of radius (a) has linear charge density $$\lambda=\lambda_0cos^2\theta$$ where (\theta) is the angle with respect to a fixed radius. Calculate the total charge.

Solution:

Charge on an element (dl=ad\theta) is $$ \lambda_0cos^2\theta.ad\theta$$
Total charge $$ Q=\int_{0}^{2\pi}a\lambda_0cos^2\theta d\theta$$$$=a\lambda_0\pi$$

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