Physics · Ch 5 — Electrostatic Potential and Capacitance

Introduction

5.1

Introduction

Why Potential Energy Exists for Charges

In earlier chapters, you learned that conservative forces (like gravity and spring force) store work as potential energy. The Coulomb force between two stationary charges is also conservative — it follows the same inverse-square law as gravity, with mass replaced by charge. This means we can define electrostatic potential energy for a charge in an electric field, just like gravitational potential energy for a mass.

The Thought Experiment: Moving a Test Charge

Consider a fixed charge $Q$ at the origin, creating an electric field $\mathbf{E}$ . We bring a small test charge $q$ (positive, and so small it doesn't disturb $Q$ ) from point $R$ to point $P$ against the repulsive electric force $F_E$ .

To move $q$ without accelerating it, we apply an external force $\mathbf{F}_{\text{ext}}$ that exactly cancels the electric force:

\mathbf{F}_{\text{ext}} = -\mathbf{F}_E

The work done by this external force, $W_{RP}$ , is stored entirely as potential energy of the charge $q$ .
If the external force is removed at $P$ , the electric force pushes $q$ away, converting that stored energy into kinetic energy — total mechanical energy is conserved.

Defining Potential Energy Difference

The work done by the external force in moving $q$ from $R$ to $P$ equals the change in electrostatic potential energy:

\Delta U = U_P - U_R = W_{RP}

Here:

$U_P$ = potential energy at point $P$
$U_R$ = potential energy at point $R$
$W_{RP}$ = work done by external force (against the electric field)

Key insight: This work depends only on the initial and final positions, not on the path taken. This path-independence is the hallmark of a conservative force — it makes the concept of potential energy meaningful.

The Arbitrary Zero Point

Potential energy itself is defined only up to an additive constant. Only differences in potential energy are physically significant. We can always choose a reference point where potential energy is zero.

Convenient choice: Set $U = 0$ at infinity ( $R = \infty$ ). Then for any point $P$ :

U_P = W_{\infty P}

That is:

The electrostatic potential energy of a charge $q$ at a point is the work done by an external force in bringing $q$ from infinity to that point, without acceleration.

Figure 2.1A test charge q (> 0) is moved from the point R to the point P against the repulsive force on it by the charge Q (> 0) placed at the origin.

Fig. 2.1 — A test charge q (> 0) is moved from the point R to the point P against the repulsive force on it by the charge Q (> 0) placed at the origin.

Drawn by us to help you understand the concept clearly, and verified to make sure it's accurate. For exams, practice from your textbook's own diagram.

The figure shows a source charge $Q$ (positive, $Q > 0$ ) fixed at the origin, represented by a circle with a plus sign ( $\oplus$ ). A straight line (the path) runs from a point labelled P (lower-left end) to a point labelled R (upper-right end). A second $\oplus$ symbol represents the test charge $q$ (also positive, $q > 0$ ) placed on this path about two-thirds of the way from P to R. A small arrowhead on the line points backward from R toward P, indicating the direction of motion: the test charge is moved from R to P.

The physical idea is that because both $Q$ and $q$ are positive, the Coulomb force on $q$ is repulsive — it pushes $q$ away from the origin. To move $q$ from R (farther from $Q$ ) to P (closer to $Q$ ), an external force $\mathbf{F}_{\text{ext}}$ must be applied, exactly opposite to the electric force $\mathbf{F}_E$ , so that $q$ moves with infinitesimally slow constant speed (no net acceleration). The work done by this external force is stored as electrostatic potential energy of the charge $q$ at point P relative to point R.

The key formula developed from this figure is the definition of potential energy difference:

\Delta U = U_P - U_R = W_{RP}

where:

$U_P$ = electrostatic potential energy of $q$ at point P,
$U_R$ = electrostatic potential energy of $q$ at point R,
$W_{RP}$ = work done by the external force in moving $q$ from R to P (against the electric force).

The text then extends this to define potential energy at a point by choosing infinity as the reference (where $U_\infty = 0$ ):

U_P = W_{\infty P}

Here $W_{\infty P}$ is the work done by the external force in bringing $q$ from infinity to point P. This work is path-independent because the electrostatic force is conservative — a fact the figure’s straight-line path illustrates, but the result holds for any path.

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