Question

A capacitor consists of two large metal disks placed at a distance apart. The radius of each disk is $\mathit{R}\left(R>>s\right)$, and the thickness of each disk is$\mathit{t}$, as shown in Figure 16.73. The disk on the left has a net charge of$\mathbf{+}\mathit{Q}$, and the disk on the right has a net charge of$\mathbf{-}\mathit{Q}$. Calculate the potential difference ${\mathit{V}}_{\mathbf{2}}\mathbf{-}{\mathit{V}}_{\mathbf{1}}$, where location 1 is inside the left disk at its center, and location 2 is in the center of the air gap between the disks. Explain briefly

Short answer

The potential difference between the points 1 and 2 is $\frac{Qs}{2{\epsilon }_0\pi R^2}$.

Step-by-step solution

Write the given data from the question.

The distance between the capacitor disks plates is $s$.

The thickness of the disk is $t$.

The radius of the disk is $R$.

The net charge on left disk is $+Q$and on right disk is $-Q$.

Determine the formulas to calculate the potential difference.

The expression to calculate the electrical field between the places is given as follows.

$\mathit{E}\mathbf{=}\frac{\mathbf{Q}}{{\mathbf{\epsilon }}_{\mathbf{0}}\mathbf{A}}$ …… (i)

Here,$A$ is the area of the disk.

The expression to calculate the potential difference between the point 1 and 2 is given as follows.

${\mathit{V}}_{\mathbf{2}}\mathbf{-}{\mathit{V}}_{\mathbf{1}}\mathbf{=}\mathit{E}\mathit{d}$…… (ii)

calculate the potential difference between the point 1 and 2.

The distance between the point 1 and 2 is $s/2$.

The area of the disk is given by,

$A=\pi R^2$

Calculate the electrical field.

Substitute $\pi R^2$for $A$into equation (i).

$E=\frac{Q}{{\epsilon }_0\pi R^2}$

Calculate the potential difference between the point 1 and 2,

Substitute $\frac{Q}{{\epsilon }_0\pi R^2}$for $E$and $s/2$for $d$into equation (ii).

$$\begin{gathered} V_2-V_1=\frac{Q}{{\epsilon }_0\pi R^2}\times \frac{s}{2} \\ {V}_2-V_1=\frac{Qs}{2{\epsilon }_0\pi R^2} \end{gathered}$$

Hence the potential difference between the points 1 and 2 is$\frac{Qs}{2{\epsilon }_0\pi R^2}$ .