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Total Charge on a Circular Disc

Let's discuss a problem where we will find out the total charge on a circular disc.

The Problem:

Consider a circular disc of radius (a) whose surface density of charge at any point ((r,\theta)) is $$ \sigma(r,\theta)=\sigma_0r^2sin^2\theta$$ Find the total charge on the disc.

Solution:

Consider a surface element (rdrd\theta) on the disc about the point ((r,\theta)). The charge on is $$Q= \sigma(r,\theta)rd\theta dr$$ $$=\sigma_0\int_{0}^{a} r^3dr \int_{0}^{2\pi}sin^2\theta d\theta$$ $$=\sigma_0\frac{a^4}{4}.\pi$$$$=\pi \sigma_0\frac{a^4}{4}$$

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