Field Inside Charged Solid Sphere

Question

For uniformly charged solid sphere (total charge Q, radius R), field at distance r<R from center?

RankGuru · Accepted Answer

1. **Gauss's Law**: $\oint E \cdot dA = Q_{enc} / \epsilon_0$.
2. **Gaussian Surface**: Pick a sphere inside radius $r < R$.
3. **Enclosed Charge**: Since volume density $ho = Q / (\frac{4}{3}\pi R^3)$, then $Q_{enc} = ho 	imes (\frac{4}{3}\pi r^3) = Q \frac{r^3}{R^3}$.
4. **Equating**: $E (4\pi r^2) = \frac{1}{\epsilon_0} \cdot Q \left(\frac{r}{R}ight)^3 R = \frac{Q r^3}{\epsilon_0 R^3}$.
5. **Solve for E**: $E = \frac{1}{4\pi \epsilon_0} \frac{Q r}{R^3} = k \frac{Q r}{R^3}$.
6. **Linearity**: The electric field inside a uniformly charged solid non-conducting sphere increases linearly with $r$ from the center.

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Field Inside Charged Solid Sphere

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