# Total Charge of a Sphere

Suppose a charge $$Q$$ is distributed within a sphere of radius $$R$$ in such a way that the charge density $$\rho(r)$$ at a distance r from the centre of the sphere is
$$\rho(r)=K(R-r) \hspace{2mm }for\hspace{2mm} 0<r<R$$
$$0 \hspace{2mm} for \hspace{2mm} r>R$$

Determine the total charge $$Q$$.
Solution:

Let us consider a thin spherical shell of radius $$r$$ and thickness $$dr$$. Charge within it is $$\rho(r).4\pi r^2dr$$. Therefore, the total charge $$Q=\int_{0}^{R}\rho(x).4\pi r^2dr$$$$=4\pi K\int_{0}^{R}(R-r)^2dr$$$$=\pi KR^4/3$$