Question 243: Electrostats & Capacitance

Question

An infinitely long thin wire, having a uniform charge density per unit length of $5 \mathrm{nC} / \mathrm{m}$, is passing through a spherical shell of radius $1 \mathrm{~m}$, as shown in the figure. A $10 \mathrm{nC}$ charge is distributed uniformly over the spherical shell. If the configuration of the charges remains static, the magnitude of the potential difference between points $\mathrm{P}$ and $\mathrm{R}$, in Volt, is ________.
[Given: In SI units $\frac{1}{4 \pi \epsilon_0}=9 	imes 10^9, \ln 2=0.7$. Ignore the area pierced by the wire.]

Accepted Answer

due to wire
$$\begin{aligned}
& \mathrm{dV}=-\overrightarrow{\mathrm{E}} \cdot \overrightarrow{\mathrm{dx}} \
& \int_\limits{\mathrm{v}_{\mathrm{P}}}^{\mathrm{v}_{\mathrm{R}}} \mathrm{dV}=-\int_\limits{0.5}^2 \frac{2 \mathrm{k} \lambda}{\mathrm{x}} \mathrm{dx} \
& \mathrm{v}_{\mathrm{R}}-\mathrm{v}_{\mathrm{P}}=-2 \mathrm{k} \lambda \ln \frac{2}{0.5} \
& =-2 	imes 9 	imes 10^9 	imes 3 	imes 10^{-9} 	imes 2 	imes 0.7=-126 \mathrm{~V}
\end{aligned}$$
due to sphere
$$\begin{aligned}
& v_R-v_P= \frac{k Q}{2}-\frac{k Q}{1}=-\frac{k Q}{2}=\frac{-9 	imes 10^9 	imes 10 	imes 10^{-9}}{2} \
&=-45 \mathrm{~V} \
& v_R-v_P=-126-45=-171 \mathrm{~V} \
& v_P-v_R= 171 \mathrm{~V}
\end{aligned}$$

Electrostats & Capacitance

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