The plates of a spherical capacitor have radii 38.0 mmand 40.0 mm(a) Calculate the capacitance. (b) What must be the plate area of a parallel-plate capacitor with the same plate separation and capacitance?

The spherical capacitors have radii $38.0\mathrm{mm}\left(\frac{{10}^{-3}\mathrm{m}}{1\mathrm{mm}}\right)=3.80\times {10}^{-2}\mathrm{m}\mathrm{and}\mathrm{b}=40.0\mathrm{mm}\left(\frac{{10}^{-3}\mathrm{m}}{1\mathrm{mm}}\right)=4.00\times {10}^{-2}\mathrm{m}$

The plate separation is the same.

The capacitance is also the same.

By using the equations 25-17 and 25-9, find the capacitance and the plate area of a parallel plate capacitor

Formulae are as follows:

$\mathbf{C}\mathbf{=}\mathbf{4}{\mathbf{\tau \tau \epsilon }}_{\mathbf{0}}\frac{\mathbf{ab}}{\mathbf{(}\mathbf{b}\mathbf{-}\mathbf{a}\mathbf{)}}$

Where C is capacitance, a and b are radiiis the permittivity of the medium.

From the equation 25-17, the capacitance of the spherical capacitor is given by,

$$\begin{gathered} \mathrm{C}=4{\mathrm{\tau \tau \epsilon }}_0\frac{\mathrm{ab}}{(\mathrm{b}-\mathrm{a})} \\ =4\times 3.142\times (8.85\times {10}^{-12}\mathrm{F}/\mathrm{m})\times \frac{(3.80\times {10}^{-2}\mathrm{m})\times (4.00\times {10}^{-3}\mathrm{m})}{\left[(4.00\times {10}^{-3}\mathrm{m})-(3.80\times {10}^{-2}\mathrm{m})\right]} \\ =8.45\times {10}^{-11}\mathrm{F}\left(\frac{1\mathrm{pF}}{{10}^{-12}\mathrm{F}}\right) \\ =84.5\mathrm{pF} \end{gathered}$$

Hence, the capacitance of the spherical capacitor is 84.5 pF.

From the equation 25-9, the plate area of a parallel plate capacitor is given by,

$$\begin{gathered} \mathrm{A}=\frac{\mathrm{C}(\mathrm{b}-\mathrm{a})}{{\mathrm{\epsilon }}_0} \\ \mathrm{Where},\mathrm{d}=(\mathrm{b}-\mathrm{a}) \\ \mathrm{A}=\frac{\left(8.45\times {10}^{-11}\mathrm{F}\right)\left[(4.00\times {10}^{-2}\mathrm{m})-(3.80\times {10}^{-2}\mathrm{m}\right]}{(8.85\times {10}^{-12}\mathrm{F}/\mathrm{m})} \\ =1.91\times {10}^{-2}{\mathrm{m}}^2 \end{gathered}$$

Hence, the plate area of a parallel plate capacitor is$1.91\times {10}^{-2}{\mathrm{m}}^2$.

Therefore, by using the formula the capacitance and the plate area of a parallel plate capacitor can be determined.