Q2.48P - Introduction to Electrodynamics

Question

An inverted hemispherical bowl of radius R carries a uniform surface charge density $\mathit{\sigma }$. Find the potential difference between the "north pole" and the center.

Step-by-Step Solution

Answer

Answer

The potential difference between the "north pole" and the center is $\frac{\mathbf{\sigma }\mathbf{R}}{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{0}}}\left(\sqrt{2}-1\right)$.

Step 1: Define functions

Given that, R is the radius of the hemispherical bowl,  is the charge density of the hemispherical bowl.

Calculate the potential at the center of hemispherical bowl.

${\mathit{V}}_{\mathbf{c}\mathbf{e}\mathbf{n}\mathbf{t}\mathbf{e}\mathbf{r}}\mathbf{=}\frac{\mathbf{\sigma }\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}{\mathbf{\int }}_{\mathbf{r}}\mathbf{-}\mathit{d}\mathit{a}$                           …… (1)

Then, ${\mathit{V}}_{\mathbf{center}}\mathbf{=}\frac{\mathbf{\sigma }\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}{\mathbf{\int }}_{}\mathbf{-}\mathit{d}\mathit{a}$

Here, $\mathbf{\int }\mathit{d}\mathit{a}$is the surface area of hemisphere. $\mathbf{\int }\mathit{d}\mathit{a}\mathbf{=}\mathbf{2}\mathit{\pi }{\mathit{R}}^{\mathbf{2}}$.

Thus, the potential at the center of hemispherical bowl is, 

$$\begin{gathered} {\mathit{V}}_{\mathbf{center}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}\frac{\mathbf{\sigma }}{\mathbf{R}}\mathbf{2}\mathit{\pi }{\mathit{R}}^{\mathbf{2}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\frac{\mathbf{\sigma R}}{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{0}}} \end{gathered}$$

${\mathit{V}}_{\mathbf{center}}\mathbf{=}\frac{\mathbf{\sigma R}}{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{0}}}$                                             ………. (2)

Step 2: Determine potential at the North Pole

Write the expression for potential at the North Pole using equation (1)

${\mathit{V}}_{\mathbf{p}\mathbf{o}\mathbf{l}\mathbf{e}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}\mathbf{\pi }{\mathbf{\epsilon }}_{\mathbf{0}}}\mathbf{\int }\frac{\mathbf{\sigma }}{\mathbf{r}}\mathit{d}\mathit{a}$

Here, it is not necessary to integrate the term with respect to $\mathit{\theta }$. Now consider the pole. According to the diagram the pole is overhead of the point of consideration. It makes the angle $\mathit{\theta }$ to 0.

Considering pole,

$\mathit{d}\mathit{a}\mathbf{=}\mathbf{2}\mathit{\pi }{\mathit{R}}^{\mathbf{2}}\mathit{s}\mathit{i}\mathit{n}\mathit{\theta }\mathit{d}\mathit{\theta }$

$$\begin{gathered} {\mathit{r}}^{\mathbf{2}}\mathbf{=}{\mathit{R}}^{\mathbf{2}}\mathbf{+}{\mathit{R}}^{\mathbf{2}}\mathbf{-}\mathbf{2}{\mathit{R}}^{\mathbf{2}}\mathbf{ }\mathit{c}\mathit{o}\mathit{s}\mathit{\theta } \\ {\mathit{r}}^{\mathbf{2}}\mathbf{=}\mathbf{2}{\mathit{R}}^{\mathbf{2}}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{cos}\mathbf{ }\mathbf{\theta }\mathbf{)} \\ \mathbf{ }\mathit{r}\mathbf{=}\mathit{R}\sqrt{\mathbf{2}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{cos\theta }\mathbf{)}} \end{gathered}$$

Therefore, the pole is calculated as,

$$\begin{gathered} {\mathit{V}}_{\mathbf{pole}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{4}{\mathbf{\pi \epsilon }}_{\mathbf{0}}}\frac{\mathbf{\sigma }\mathbf{\left(}\mathbf{2}{\mathbf{\pi R}}^{\mathbf{2}}\mathbf{\right)}}{\mathbf{R}\sqrt{\mathbf{2}}}\underset{\mathbf{0}}{\overset{\mathbf{\pi }\mathbf{/}\mathbf{2}}{\mathbf{\int }}}\frac{\mathbf{sin\theta d\theta }}{\sqrt{\mathbf{1}\mathbf{-}\mathbf{cos\theta }}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\frac{\mathbf{\sigma R}}{\sqrt{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{0}}}}{\mathbf{\left(}\mathbf{2}\sqrt{\mathbf{1}\mathbf{-}\mathbf{cos\theta }}\mathbf{\right)}}_{\mathbf{0}}^{\mathbf{\pi }\mathbf{/}\mathbf{2}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\frac{\mathbf{\sigma R}}{\sqrt{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{0}}}}\mathbf{(}\mathbf{1}\mathbf{-}\mathbf{0}\mathbf{)} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\frac{\mathbf{\sigma R}}{\sqrt{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{0}}}} \end{gathered}$$

Therefore, the north pole is $\frac{\mathbf{\sigma R}}{\sqrt{\mathbf{2}}{\mathbf{\epsilon }}_{\mathbf{0}}}$.

Step 3: Determine potential difference between the North Pole and center

$$\begin{gathered} {\mathit{V}}_{\mathbf{p}\mathbf{o}\mathbf{l}\mathbf{e}}\mathbf{-}{\mathit{V}}_{\mathbf{p}\mathbf{o}\mathbf{l}\mathbf{e}}\mathbf{=}\frac{\mathbf{\sigma }\mathbf{R}}{\sqrt{\mathbf{2}}{\mathbf{\epsilon }}_{\mathbf{0}}}\mathbf{-}\frac{\mathbf{\sigma }\mathbf{R}}{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{0}}} \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\frac{\mathbf{\sigma }\mathbf{R}}{\sqrt{\mathbf{2}}{\mathbf{\epsilon }}_{\mathbf{0}}}\left[1-\frac{1}{\sqrt{2}}\right] \\ \mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{=}\frac{\mathbf{\sigma R}}{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{0}}}\mathbf{(}\sqrt{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{)} \end{gathered}$$

Hence, the potential difference between the "north pole" and the center is $\frac{\mathbf{\sigma R}}{\mathbf{2}{\mathbf{\epsilon }}_{\mathbf{0}}}\mathbf{(}\sqrt{\mathbf{2}}\mathbf{-}\mathbf{1}\mathbf{)}$.