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Chapter 2: Problem 30

A spherical capacitor has an inner sphere of radius \(12 \mathrm{~cm}\) and an outer sphere of radius \(13 \mathrm{~cm} .\) The outer sphere is earthed and the inner sphere is given a charge of \(2.5 \mu \mathrm{C}\). The space between the concentric spheres is filled with a liquid of dielectric constant 32 . (a) Determine the capacitance of the capacitor. (b) What is the potential of the inner sphere? (c) Compare the capacitance of this capacitor with that of an isolated sphere of radius \(12 \mathrm{~cm}\). Explain why the latter is much smaller.

Short Answer

Expert verified
The capacitance of the spherical capacitor in question is significantly larger than that of an isolated sphere of the same radius due to the presence of a dielectric medium. The polarization in the dielectric contributes to a decrease in the potential difference, resulting in a larger capacitance.

Step by step solution

01

Calculating capacitance

Use the formula for the capacitance of a spherical capacitor: \(C= 4蟺蔚蔚^' \frac{a*b}{b-a}\), substituting the given values, i.e., \(a = 12*10^{-2} m, b = 13*10^{-2} m, 蔚' =32\), and \(蔚 = 8.85*10^{-12} F/m\). By inserting these values the capacitance can be calculated.
02

Calculating the potential of the inner sphere

Use the formula for potential (\(V=\frac{Q}{C}\)), substituting the given charge Q \(=2.5*10^{-6} C\) and the calculated capacitance C from Step 1, to find the potential of the inner sphere.
03

Comparing the capacitances

The capacitance of an isolated sphere is given by the formula \(C= 4蟺蔚R\), where R is the radius of the sphere. Now, compare this capacitance with the capacitance calculated in Step 1 and explain why the latter is much smaller.
04

Explaining the difference

The significant difference in capacitance arises because the capacitance of the spherical capacitor including the dielectric is greater due to the polarization of the dielectric medium. When a dielectric is present, the net charge in the capacitor decreases, causing the potential difference to decrease, thereby increasing the capacitance. This phenomenon is not present in an isolated spherical capacitor that doesn't have a dielectric medium.

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Most popular questions from this chapter

Two charged conducting spheres of radii \(a\) and \(b\) are connected to each other by a wire. What is the ratio of electric fields at the surfaces of the two spheres? Use the result obtained to explain why charge density on the sharp and pointed ends of a conductor is higher than on its flatter portions.

A spherical capacitor has an inner sphere of radius \(12 \mathrm{~cm}\) and an outer sphere of radius \(13 \mathrm{~cm} .\) The outer sphere is earthed and the inner sphere is given a charge of \(2.5 \mu \mathrm{C}\). The space between the concentric spheres is filled with a liquid of dielectric constant 32 . (a) Determine the capacitance of the capacitor. (b) What is the potential of the inner sphere? (c) Compare the capacitance of this capacitor with that of an isolated sphere of radius \(12 \mathrm{~cm}\). Explain why the latter is much smaller.

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