Question

A spherical capacitor is made from two thin concentric conducting shells. The inner shell has radius \(r_{1}\), and the outer shell has radius \(r_{2}\). What is the fractional difference in the capacitances of this spherical capacitor and a parallel plate capacitor made from plates that have the same area as the inner sphere and the same separation \(d=r_{2}-r_{1}\) between them?

Short answer

Question: Determine the fractional difference in capacitances between a spherical capacitor and a parallel plate capacitor, given the radii of the concentric conducting shells of the spherical capacitor as \(r_1\) and \(r_2\), and the separation between the plates of the parallel plate capacitor as \(d = r_2 - r_1\). Answer: The fractional difference in capacitances between the spherical capacitor and the parallel plate capacitor is \(\frac{r_2 - r_1}{r_1}\).

Step-by-step solution

Express the capacitance of a spherical capacitor

The capacitance of a spherical capacitor is given by the formula: $$C_s = 4 \pi \epsilon_0 \frac{r_1 r_2}{r_2 - r_1}$$ where \(\epsilon_0\) is the vacuum permittivity.

Express the capacitance of a parallel plate capacitor

The capacitance of a parallel plate capacitor is given by the formula: $$C_p = \frac{\epsilon_0 A}{d}$$ where \(A\) is the area of the plates and \(d\) is the separation between them. In our case, \(A = 4 \pi r_1^2\) (the surface area of the inner sphere) and \(d = r_2 - r_1\).

Calculate the capacitance of the parallel plate capacitor

Plugging in the values for \(A\) and \(d\), we get: $$C_p = \frac{\epsilon_0 (4 \pi r_1^2)}{r_2 - r_1}$$

Determine the fractional difference in capacitances

Now, we need to find the fractional difference between the capacitances of the spherical capacitor and the parallel plate capacitor. We can calculate this as: $$\text{Fractional difference} = \frac{C_s - C_p}{C_p}$$ Plugging in the values for \(C_s\) and \(C_p\), we get: $$\text{Fractional difference} = \frac{4 \pi \epsilon_0 \frac{r_1 r_2}{r_2 - r_1} - \frac{\epsilon_0 (4 \pi r_1^2)}{r_2 - r_1}}{\frac{\epsilon_0 (4 \pi r_1^2)}{r_2 - r_1}}$$ Now, we can simplify the expression: $$\text{Fractional difference} = \frac{4 \pi r_1 r_2 - 4 \pi r_1^2}{4 \pi r_1^2}$$ $$\text{Fractional difference} = \frac{4 \pi r_1 (r_2 - r_1)}{4 \pi r_1^2}$$ Cancelling out the common terms, we get the final expression for the fractional difference: $$\text{Fractional difference} = \frac{r_2 - r_1}{r_1}$$