Question 783: Electrostats & Capacitance

Question

Three concentric charged metallic spherical sheets $A, B$ and $C$ have radii $a, b, c$ potentials $V_A$, $\mathrm{V}_{\mathrm{B}}, \mathrm{V}_{\mathrm{C}}$ and charge densities $+\sigma,-\sigma$ and $+\sigma$ respectively. The value of potential $\mathrm{V}_{\mathrm{A}}$ is ( $\varepsilon_0=$ permittivity of free space)

Accepted Answer

Potential and conducing shell = sum of potentials due to all other shells (including self) at its own radius.
i. Potential due to A on its own surface.
$$ \mathrm{V}_{\mathrm{A} ightarrow \mathrm{~A}}=\frac{1}{4 \pi \varepsilon_0} \frac{\mathrm{Q}_{\mathrm{A}}}{\mathrm{a}}=\frac{1}{4 \pi \varepsilon_0} \frac{4 \pi \mathrm{a}^2 \sigma}{\mathrm{a}}=\frac{\mathrm{a} \sigma}{\varepsilon_0} $$
ii. Potential at A due to shell B
$$ \mathrm{V}_{\mathrm{B} ightarrow \mathrm{~A}}=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{\mathrm{Q}_{\mathrm{B}}}{\mathrm{~b}}=\frac{1}{4 \pi \varepsilon_0} \frac{4 \pi \mathrm{~b}^2 \sigma}{\mathrm{~b}}=\frac{-\mathrm{b} \sigma}{\varepsilon_0} $$
iii. Potential at A due to shell C
$$ \begin{aligned} & V_{C ightarrow A}=\frac{1}{4 \pi \varepsilon_0} \cdot \frac{Q_C}{c}=\frac{1}{4 \pi \varepsilon_0} \frac{4 \pi c^2 \sigma}{c}=\frac{c \sigma}{\varepsilon_0} \ 	herefore \quad V_A & =V_{A ightarrow A}+V_{B ightarrow A}+V_{C ightarrow A} \ & =\frac{a \sigma}{\varepsilon_0}-\frac{b \sigma}{\varepsilon_0}+\frac{c \sigma}{\varepsilon_0} \ V_A & =\frac{\sigma}{\varepsilon_0}(a-b+c) \end{aligned} $$

Answer

$\quad \frac{\sigma}{\varepsilon_0}(\mathrm{a}+\mathrm{b}+\mathrm{c})$

Answer

$\frac{\sigma}{\varepsilon_0}(-\mathrm{a}+\mathrm{b}-\mathrm{c})$

Answer

$\frac{\sigma}{\varepsilon_0}(\mathrm{a}-\mathrm{b}+\mathrm{c})$

Answer

$\quad \frac{\sigma}{3 \varepsilon_0}(\mathrm{a}+\mathrm{b}+\mathrm{c})$

Electrostats & Capacitance

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