Consider a sphere of radius $R$ with charge density distributed as :
$\rho(r) =k r$, $r \leq R $
$=0$ for $r> R$.
$(a)$ Find the electric field at all points $r$.
$(b)$ Suppose the total charge on the sphere is $2e$ where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.
Consider a sphere of radius $R$ with charge density distributed as :
$\rho(r) =k r$, $r \leq R $
$=0$ for $r> R$.
$(a)$ Find the electric field at all points $r$.
$(b)$ Suppose the total charge on the sphere is $2e$ where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.
$(a)$ Let us consider a sphere $\mathrm{S}$ of radius $\mathrm{R}$ and two hypothetic spheres of radius $r<\mathrm{R}$ and $r>\mathrm{R}$. Electric field intensity for point $r<\mathrm{R}$,
$\oint \overrightarrow{\mathrm{E}} \cdot d \overrightarrow{\mathrm{S}}=\frac{1}{\epsilon_{0}} \int \rho d \mathrm{~V}$
But, volume $\mathrm{V}=\frac{4}{3} \pi r^{3}$
$d \mathrm{~V} =\frac{4}{3} \pi \times 3 r^{2} d r$
$d \mathrm{~V} =4 \pi r^{2} d r$
$\therefore \oint \overrightarrow{\mathrm{E}} \cdot d \overrightarrow{\mathrm{S}}=\frac{1}{\epsilon_{0}} 4 \pi k \int_{0}^{r} r^{3} d r \quad[\rho=k r]$ $\therefore(\mathrm{E}) 4 \pi r^{2}=\frac{4 \pi k}{\epsilon_{0}} \cdot \frac{r^{4}}{4}$ $\therefore \mathrm{E}=\frac{1}{4 \epsilon_{0}} \cdot k r^{2}$ As charge density is positive it means the direction of $\overrightarrow{\mathrm{E}}$ is radially outwards.
Similar Questions
At a point $20\, cm$ from the centre of a uniformly charged dielectric sphere of radius $10\, cm$, the electric field is $100\, V/m$. The electric field at $3\, cm$ from the centre of the sphere will be.......$V/m$
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A spherically symmetric charge distribution is considered with charge density varying as
$\rho(r)=\left\{\begin{array}{ll}\rho_{0}\left(\frac{3}{4}-\frac{r}{R}\right) & \text { for } r \leq R \\ \text { Zero } & \text { for } r>R\end{array}\right.$
Where, $r ( r < R )$ is the distance from the centre $O$ (as shown in figure). The electric field at point $P$ will be.
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$\rho(r)=\left\{\begin{array}{ll}\rho_{0}\left(\frac{3}{4}-\frac{r}{R}\right) & \text { for } r \leq R \\ \text { Zero } & \text { for } r>R\end{array}\right.$
Where, $r ( r < R )$ is the distance from the centre $O$ (as shown in figure). The electric field at point $P$ will be.
- [JEE MAIN 2022]
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