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Chapter 23: Problem 6

A student argues that the total energy associated with the electric field of a charged sphere must be infinite because its field extends throughout an infinite volume. Critique this argument.

Short Answer

Expert verified
The student's argument is incorrect because although the electric field extends to infinity, the energy of the electric field is not solely determined by the volume it covers. More specifically, the energy density decreases with distance from the source. Integration reveals that the energy is not infinite but rather finite and directly related to the charge and radius of the sphere.

Step by step solution

01

Understand the Basic Concepts

The energy of an electric field isn't dictated solely by its extent. It also depends on the intensity of the field, which is defined by the charge and distance from the source according to Coulomb's law. As per this law, the electric field decreases as the distance from the sphere increases. Therefore, the energy density gets diluted as we move farther from the sphere.
02

Calculate Energy Density

The energy density \(U\) associated with an electric field \(E\) in free space is given by \(U = 1/2 \epsilon E^2\), where \(\epsilon\) is the permittivity of free space. For a sphere of radius \(R\) and uniform charge \(Q\), the field is given by \(E = Q/ 4\pi\epsilon r^2\) for \(r \geq R\) .
03

Integrate to Find Total Energy

The total energy \(W\) in the electric field surrounding the sphere can be found by integrating the energy density over the infinite volume around the sphere. Performing this integration, the total energy is given by \(W = Q^2 / 8\pi\epsilon R\). This shows the energy is finite, contradicting the student's argument.

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Most popular questions from this chapter

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?

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