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Chapter 23: Problem 2

How does the energy density at a certain distance from a negative point charge compare with the energy density at the same distance from a positive point charge of equal magnitude?

Short Answer

Expert verified
The energy density at a certain distance from a negative point charge is equal to the energy density at the same distance from a positive point charge of equal magnitude. This is because energy density depends on the square of the charge magnitude and not the sign of the charge.

Step by step solution

01

Define electric field intensity

The electric field intensity \(E\) at a distance \(r\) from a point charge \(Q\) is given by Coulomb's Law: \(E = \frac{kQ}{r^2}\) where \(k\) is Coulomb's constant.
02

Define energy density

The energy density \(u\) at a particular point in an electric field is given by the formula: \(u = \frac{1}{2} ε E^2\), where \(ε\) is the permittivity of free space and \(E\) is the electric field intensity at that point.
03

Express energy density in terms of point charge and distance

Substitute the expression for electric field intensity into the formula of energy density: \(u = \frac{1}{2} ε (\frac{kQ}{r^2})^2\). Simplify to get \(u = \frac{1}{2} ε \frac{k^2 Q^2}{r^4}\). Because the energy density is proportional to the square of the charge magnitude \(Q^2\), it does not depend on the sign of the charge.
04

Compare energy densities

Since the energy density \(u\) depends solely on \(Q^2\) (the square of the charge magnitude) and not on the sign of \(Q\), the energy densities at a certain distance from a negative point charge and from a positive point charge of the same magnitude are equal. The sign of the charge does not affect the energy density at a given distance.

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