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Chapter 24: Problem 28

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 plates?

Short Answer

Expert verified
Answer: The fractional difference in the capacitances of the spherical capacitor and the parallel plate capacitor is given by \(\frac{r_2 - r_1}{r_1}\), where \(r_1\) and \(r_2\) are the radii of the inner and outer spheres of the spherical capacitor, respectively.

Step by step solution

01

Capacitance of a Spherical Capacitor

To calculate the capacitance of a spherical capacitor made from two thin concentric conducting shells with radii \(r_{1}\) and \(r_{2}\), we use the following formula: \(C_\text{spherical} = 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1}\) where \(\varepsilon_0\) is the vacuum permittivity.
02

Capacitance of a Parallel Plate Capacitor

To calculate the capacitance of a parallel plate capacitor with the same area as the inner sphere and the same separation \(d = r_2 - r_1\) between the plates, we use the following formula: \(C_\text{parallel} = \frac{\varepsilon_0 A}{d}\) where \(A\) is the area of the plates which is equal to the surface area of the inner sphere: \(A = 4 \pi r_1^2\) Therefore, the capacitance of the parallel plate capacitor is: \(C_\text{parallel} = \frac{\varepsilon_0 \cdot 4 \pi r_1^2}{r_2 - r_1}\)
03

Find the Difference in Capacitances

We will now find the difference between the capacitances of the spherical and parallel plate capacitors: \(\Delta C = C_\text{spherical} - C_\text{parallel}\) Substituting the values, we get: \(\Delta C = 4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1} - \frac{\varepsilon_0 \cdot 4 \pi r_1^2}{r_2 - r_1}\)
04

Calculate the Fractional Difference

Finally, we will calculate the fractional difference between the capacitances. The fractional difference is given by: \(\text{Fractional Difference} = \frac{\Delta C}{C_\text{parallel}}\) Substituting the values, we get: \(\text{Fractional Difference} = \frac{4 \pi \varepsilon_0 \frac{r_1 r_2}{r_2 - r_1} - \frac{\varepsilon_0 \cdot 4 \pi r_1^2}{r_2 - r_1}}{\frac{\varepsilon_0 \cdot 4 \pi r_1^2}{r_2 - r_1}}\) Simplifying, we get: \(\text{Fractional Difference} = \frac{r_2 - r_1}{r_1}\) This is the fractional difference in the capacitances of the spherical capacitor and the parallel plate capacitor.

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Most popular questions from this chapter

How much energy can be stored in a capacitor with two parallel plates, each with an area of \(64.0 \mathrm{~cm}^{2}\) and separated by a gap of \(1.30 \mathrm{~mm}\), filled with porcelain whose dielectric constant is \(7.0,\) and holding equal and opposite charges of magnitude \(420 . \mu C ?\)

Which of the following capacitors has the largest charge? a) a parallel plate capacitor with an area of \(10 \mathrm{~cm}^{2}\) and a plate separation of \(2 \mathrm{~mm}\) connected to a \(10-\mathrm{V}\) battery b) a parallel plate capacitor with an area of \(5 \mathrm{~cm}^{2}\) and a plate separation of \(1 \mathrm{~mm}\) connected to a \(10-\mathrm{V}\) battery c) a parallel plate capacitor with an area of \(10 \mathrm{~cm}^{2}\) and a plate separation of \(4 \mathrm{~mm}\) connected to a \(5-\mathrm{V}\) battery d) a parallel plate capacitor with an area of \(20 \mathrm{~cm}^{2}\) and a plate separation of \(2 \mathrm{~mm}\) connected to a \(20-\mathrm{V}\) battery e) All of the capacitors have the same charge.

The space between the plates of an isolated parallel plate capacitor is filled with a slab of dielectric material. The magnitude of the charge \(Q\) on each plate is kept constant. If the dielectric material is removed from between the plates, the energy stored in the capacitor a) increases. c) decreases. b) stays the same. d) may increase or decrease.

The distance between the plates of a parallel plate capacitor is reduced by half and the area of the plates is doubled. What happens to the capacitance? a) It remains unchanged. b) It doubles. c) It quadruples. d) It is reduced by half.

Two concentric metal spheres are found to have a potential difference of \(900 . \mathrm{V}\) when a charge of \(6.726 \cdot 10^{-8} \mathrm{C}\) is applied to them. The radius of the outer sphere is \(0.210 \mathrm{~m}\). What is the radius of the inner sphere?
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