Question 847: Electrostatics & Capacitance

Question

A charge $q$ is surrounded by a closed surface consisting of an inverted cone of height $h$ and base radius $R$, and a hemisphere of radius $R$ as shown in the figure. The electric flux through the conical surface is $\frac{n q}{6 \epsilon_{0}}$ (in SI units). The value of $n$ is _______.

Accepted Answer

Sol. From Gauss law,
$$
\phi_{	ext {hemisphere }}+\phi_{	ext {Cone }}=\frac{\mathrm{q}}{\varepsilon_0}
$$ ............(1)
Total flux produced from $\mathrm{q}$ in $\alpha$ angle
$$
\phi=\frac{\mathrm{q}}{2 \varepsilon_0}[1-\cos \alpha]
$$
For hemisphere, $\alpha=\frac{\pi}{2}$
$$
\phi_{	ext {hemisphere }}=\frac{\mathrm{q}}{2 \varepsilon_0}
$$
From equation (1)
$$
\begin{aligned}
& =\frac{\mathrm{q}}{2 \varepsilon_0}+\phi_{	ext {cone }}=\frac{\mathrm{q}}{\varepsilon_0} \\
& \phi_{	ext {cone }}=\frac{\mathrm{q}}{2 \varepsilon_0} \\
& \frac{4 \mathrm{q}}{6 \varepsilon_0}=\frac{\mathrm{q}}{2 \varepsilon_0} \\
& \mathrm{n}=3
\end{aligned}
$$

Electrostatics & Capacitance

A charge q is surrounded by a closed surface consisting of an inverted cone of height h and base radius R, and a hemisphere of radius R as shown in the figure. The electric flux through the conical surface is \frac{n q}{6 \epsilon_{0}} (in SI units). The value of n is _______.

Integer / Numeric Answer Type