CUET Preparation Today
Let $ρ(r)=\frac{Q}{\pi R^4}$ be the charge density distribution for a solid non-conducting sphere of radius R and total charge Q. For a point P inside the sphere at distance r, from the centre of the sphere, the magnitude of electric field is |
zero $\frac{Q}{4πε_0r_1^2}$ $\frac{Qr_1^2}{4πε_0R^4}$ $\frac{Qr_1^2}{3πε_0R^4}$ |
$\frac{Qr_1^2}{4πε_0R^4}$ |
The correct answer is option 3: $\frac{Qr_1^2}{4πε_0R^4}$ $$q_{\text{enc}} = \int_{0}^{r} \left( \frac{Q}{\pi R^4}r \right) \cdot 4\pi r^2 dr = \frac{4Q}{R^4} \int_{0}^{r} r^3 dr = \frac{Qr^4}{R^4}$$
Now, applying Gauss's Law: $$E(4\pi r^2) = \frac{q_{\text{enc}}}{\varepsilon_0} = \frac{Qr^4}{\varepsilon_0 R^4}$$
$$E = \frac{Qr^4}{4\pi\varepsilon_0 R^4 r^2} = \frac{Qr^2}{4\pi\varepsilon_0 R^4}$$
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