A sphere of radius R has a volume charge dens | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) According to Gauss's Law, the electric flux through a spherical surface of radius $R$ is given by $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enclose

Vedclass · Accepted Answer

$\frac{k R^{2}}{4 \varepsilon_{0}}$

Vedclass · Answer

(B) According to Gauss's Law, the electric flux through a spherical surface of radius $R$ is given by $\oint \vec{E} \cdot d\vec{A} = \frac{q_{enclosed}}{\varepsilon_{0}}$.For a sphere of radius $R$, the electric field $E$ at the surface is uniform, so $E(4 \pi R^{2}) = \frac{q_{enclosed}}{\varepsilon_{0}}$.The total charge $q_{enclosed}$ is calculated by integrating the volume charge density $\rho = k r$ over the volume of the sphere:$q_{enclosed} = \int_{0}^{R} \rho (4 \pi r^{2}) dr = \int_{0}^{R} (k r) (4 \pi r^{2}) dr = 4 \pi k \int_{0}^{R} r^{3} dr = 4 \pi k \left[ \frac{r^{4}}{4} \right]_{0}^{R} = \pi k R^{4}$.Substituting this into Gauss's Law:$E(4 \pi R^{2}) = \frac{\pi k R^{4}}{\varepsilon_{0}}$.Solving for $E$:$E = \frac{\pi k R^{4}}{4 \pi R^{2} \varepsilon_{0}} = \frac{k R^{2}}{4 \varepsilon_{0}}$.

$A$ sphere of radius $R$ has a volume charge density $\rho = k r$, where $r$ is the distance from the center of the sphere and $k$ is a constant. The magnitude of the electric field at the surface of the sphere is given by ($\varepsilon_{0} =$ permittivity of free space):

Similar Questions

$A$ solid metal sphere of radius $R$ having charge $q$ is enclosed inside a concentric spherical shell of inner radius $a$ and outer radius $b$ as shown in the figure. The approximate variation of the electric field $\overrightarrow{E}$ as a function of distance $r$ from the centre $O$ is given by:

The electric field due to an infinitely long thin straight wire with uniform linear charge density of $2.5 \times 10^{-7} \ Cm^{-1}$ at a radial distance of $x$ from the wire is $7.5 \times 10^4 \ NC^{-1}$. Then $x=$ (in $cm$)

Which graph shows the variation of the electric field of a uniformly charged non-conducting sphere with respect to the distance $(r)$ from the centre?

The electric intensity outside a charged sphere of radius $R$ at a distance $r$ $(r > R)$ is

Find the ratio of the electric field at points $A$ and $B$. An infinitely long uniformly charged wire with linear charge density $\lambda$ is kept along the $z$-axis.

Vedclass Test Series

Exam Paper Generator

Online Exam Module