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 $$ Q=\int_{0}^{R}\rho(x).4\pi r^2dr$$$$=4\pi K\int_{0}^{R}(R-r)^2dr$$$$=\pi KR^4/3$$

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