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Chapter 23: Problem 61

A sphere of radius \(R\) carries total charge \(Q\) distributed uniformly over its surface. Show that the energy stored in its electric field is \(U=k Q^{2} / 2 R\).

Short Answer

Expert verified
The energy stored in the electric field of a sphere of radius R carrying total charge Q distributed uniformly over its surface is given by \(U = k Q^{2} / 2R\)

Step by step solution

01

Understand the basics

The energy stored in an electric field is given by \(E = \frac{1}{2} \int E^2 dv\), where E is the electric field and dv is an infinitesimal volume element over which we integrate. For a surface charge, \(E = k \cdot \frac{Q}{r^2}\), where k is a coulomb's constant, Q is the charge and r is the distance from the center of sphere.
02

Setup the energy integration

Notice that the energy will be different for the region inside and outside the sphere. So, when we setup the integral for energy, we will have two integrals, one from 0 to R and other from R to infinity. The energy is then given by the sum of these two integrals. \(E = \frac{1}{2} \int_{0}^{R} k^2 \cdot \frac{Q^2}{r^4} \cdot 4\pi r^2 dr + \frac{1}{2} \int_{R}^{\infty} k^2 \cdot \frac{Q^2}{r^4} \cdot 4\pi r^2 dr\) Note that we multiplied by \(4\pi r^2\) because that is the surface area element for a spherical shell at radius r.
03

Solve the integrals

Solving these integrals, we get: \(\frac{1}{2} \int_{0}^{R} k^2 \cdot \frac{Q^2}{r^2} dr + \frac{1}{2} \int_{R}^{\infty} k^2 \cdot \frac{Q^2}{r^2} dr\) which simplifies to \(kQ^2 (1 - 1/R)\) for the first and \(kQ^2/R\) for the second.
04

Combine the results

Adding these results, we get the total energy \(U = kQ^2 (1 - 1/R) + kQ^2/R = k Q^{2}/ 2R\). So, the energy stored in its electric field is \(U = k Q^{2} / 2R\).

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Most popular questions from this chapter

Consider a proton to be a uniformly charged sphere 1 fm in radius. Find the electric energy density at the proton's surface.

The power supply in a stereo receiver contains a \(2500-\mu \mathrm{F}\) capacitor charged to 35 V. How much energy does it store?

(a) Write the electrostatic potential energy of a pair of oppositely charged, closely spaced parallel plates as a function of their separation \(x,\) their area \(A,\) and the charge magnitude \(Q .\) (b) Differentiate with respect to \(x\) to find the magnitude of the attractive force between the plates. Why isn't the force equal to the charge on one plate times the electric field between the plates?

A solid conducting slab is inserted between the plates of a capacitor, not touching either plate. Does the capacitance increase, decrease, or remain the same?

An unknown capacitor \(C\) is connected in series with a \(3.0-\mu \mathrm{F}\) capacitor; this pair is placed in parallel with a 1.0 - \(\mu\) F capacitor, and the entire combination is put in series with a \(2.0-\mu \mathrm{F}\) capacitor. (a) Make a circuit diagram of this network. (b) When a potential difference of \(100 \mathrm{V}\) is applied across the open ends of the network, the total energy stored in all the capacitors is \(5.8 \mathrm{mJ} .\) Find \(C\).
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