ELECTRIC POTENTIAL ENERGY

The concepts of work, potential energy, and conservation of energy proved to be extremely useful in our study of mechanics. In this section we’ll show that these concepts are just as useful for understanding and analyzing electrical interactions.

Fig.1: The work done on a baseball moving in a uniform gravitational field.

Let’s begin by reviewing three essential points from Chapters 6 and 7. First, when a force acts on a particle that moves from point a to point b, the work $W_{a→b}$ done by the force is given by a line integral:

$W_{a→b}=\int_{a}^{b} \mathbf{F}.d\mathbf{l}=\int_{a}^{b} F \ cos \phi \ dl$

where $d\mathbf{l}$ is an infinitesimal displacement along the particle’s path and $\phi$ is the angle between $\mathbf{F}$ and $d\mathbf{l}$ at each point along the path.

Second, if the force $\mathbf{F}$ is conservative, as we defined the term in Section 7.3, the work done by $\mathbf{F}$ can always be expressed in terms of a potential energy U. When the particle moves from a point where the potential energy is $U_a$ to a point where it is $U_b$ the change in potential energy is $\Delta U=U_b-U_a$ and the work $W_{a→b}$ done by the force is

$W_{a→b}=U_a-U_b=-(U_b-U_a)=-\Delta U$

When $W_{a→b}$ is positive, $U_a$ is greater than $U_b$ is negative, and the potential energy decreases. That’s what happens when a baseball falls from a high point (a) to a lower point (b) under the influence of the earth’s gravity; the force of gravity does positive work, and the gravitational potential energy decreases (Fig. 1). When a tossed ball is moving upward, the gravitational force does negative work during the ascent, and the potential energy increases.

Third, the work–energy theorem says that the change in kinetic energy $\Delta K=K_b-K_a$ during a displacement equals the total work done on the particle. If only conservative forces do work, then Eq. (23.2) gives the total work, and $K_b-K_a=-(U_b-U_a)$ We usually write this as

$K_a+U_a=K_b+U_b$

That is, the total mechanical energy (kinetic plus potential) is conserved under these circumstances.

  1. Electric Potential Energy in a Uniform Field
  2. Electric Potential Energy of Two Point Charges
  3. Electric Potential Energy with Several Point Charges
  4. Potential Energy of A Single Charge
  5. Potential Energy of A Dipole In An External Field

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