Path Independence of Work Done by Coulomb Force

In the study of electromagnetism, the Coulomb force serves as the fundamental interaction between stationary electric charges. Characterized as a central force, the magnitude of this interaction is inversely proportional to the square of the distance between the charges, and its direction always lies along the straight line connecting them.

When a test charge moves through an electric field generated by another charge, the Coulomb force performs work on it. By definition, work is the cumulative effect of a force acting through a displacement. Mathematically, the work $W$ done by the Coulomb force is expressed as the line integral:

$$W = \int_{A}^{B} \vec{F} \cdot d\vec{r}$$

where $\vec{F}$ represents the Coulomb force vector and $d\vec{r}$ is the infinitesimal displacement vector. Understanding the nature of this work is essential, as it dictates not only the kinetic energy of the moving charge but also the fundamental energy dynamics of the electrostatic system.

Mathematical Proof of Path Independence

To demonstrate that the work done by the Coulomb force is independent of the path taken, we can employ a point-charge model. Consider a fixed source charge $Q$ located at the origin. We move a test charge $q$ from an initial position $A$ (at a distance $r_1$ from the origin) to a final position $B$ (at a distance $r_2$).

According to Coulomb's Law, the magnitude of the force exerted on $q$ is:

$$F = k \frac{Qq}{r^2}$$

where $k$ is the electrostatic constant.

During the displacement, the total infinitesimal displacement $d\vec{r}$ can be decomposed into a radial component ($dr$) and a tangential component. Because the Coulomb force $\vec{F}$ is strictly radial, the dot product $\vec{F} \cdot d\vec{r}$ eliminates the tangential component, meaning only radial movement contributes to the work. The integral simplifies to:

$$W_{A \to B} = \int_{r_1}^{r_2} k \frac{Qq}{r^2} dr$$

Performing the integration:

$$W_{A \to B} = kQq \int_{r_1}^{r_2} \frac{1}{r^2} dr = kQq \left[ -\frac{1}{r} \right]_{r_1}^{r_2} = kQq \left( \frac{1}{r_1} - \frac{1}{r_2} \right)$$

The resulting expression for $W_{A \to B}$ depends exclusively on the initial radius $r_1$ and the final radius $r_2$. It is entirely unaffected by the specific trajectory—whether the charge moves in a straight line, a complex curve, or a spiral. This mathematical outcome confirms that the work is path-independent.

Conservative Forces and Their Implications

The property of path independence identifies the Coulomb force as a conservative force. This classification carries profound physical implications.

1. Key Characteristics of Conservative Forces

A force that is conservative, such as the Coulomb force, exhibits two equivalent properties:

Endpoint Dependence: The work performed depends solely on the starting and ending coordinates of the charge.
Zero Work over Closed Loops: If a charge travels along any arbitrary closed path and returns to its starting point, the net work done by the force is zero: $\oint \vec{F} \cdot d\vec{r} = 0$.

2. Comparison with Non-Conservative Forces

To contextualize this, consider friction, a classic non-conservative force. The work done by friction is highly dependent on the path length; the longer the route, the more energy is dissipated as heat. In contrast, the Coulomb force does not "lose" energy to the path; instead, it facilitates a reversible exchange between kinetic energy and potential energy within the system.

From Work to Electric Potential Energy and Potential

The path independence of the Coulomb force allows us to move away from complex vector calculus and instead utilize scalar functions to describe the state of a charge within an electric field.

Electric Potential Energy

We can define Electric Potential Energy ($U$) as the negative of the work done by the Coulomb force when moving a charge from a reference point (usually infinity, where $U=0$) to a specific position $r$:

$$U(r) = - \int_{\infty}^{r} \vec{F} \cdot d\vec{r} = k \frac{Qq}{r}$$

Consequently, the work done by the force is directly related to the change in potential energy ($\Delta U$):

$$W_{A \to B} = -\Delta U = U_A - U_B$$

The Concept of Electric Potential

To further streamline analysis, we define the Electric Potential ($\phi$), which represents the potential energy per unit charge:

$$\phi = \frac{U}{q} = k \frac{Q}{r}$$

This shift is transformative for physics and engineering. Instead of performing difficult vector integrations, we can calculate the work done by simply finding the difference in potential (voltage) between two points:

$$W_{A \to B} = q(\phi_A - \phi_B)$$

Illustrative Example

Scenario:
A fixed point charge $Q = +2\mu\text{C}$ is at the origin. A test charge $q = +1\mu\text{C}$ is moved from point $A$ ($r_1 = 0.1\text{m}$) to point $B$ ($r_2 = 0.3\text{m}$).

Method 1: Direct Integration
$$W = kQq \left( \frac{1}{r_1} - \frac{1}{r_2} \right)$$
$$W \approx (8.99 \times 10^9) \cdot (2 \times 10^{-6}) \cdot (1 \times 10^{-6}) \cdot \left( \frac{1}{0.1} - \frac{1}{0.3} \right)$$
$$W \approx 17.98 \cdot (10 - 3.33) \approx 119.87 \text{ J}$$

Method 2: Potential Difference
First, calculate the potential at both points:

$\phi_A = k \frac{Q}{r_1} = 8.99 \times 10^9 \cdot \frac{2 \times 10^{-6}}{0.1} = 179,800 \text{ V}$
$\phi_B = k \frac{Q}{r_2} = 8.99 \times 10^9 \cdot \frac{2 \times 10^{-6}}{0.3} = 59,933 \text{ V}$

Then, calculate the work:
$$W = q(\phi_A - \phi_B) = 1 \times 10^{-6} \cdot (179,800 - 59,933) \approx 119.87 \text{ J}$$

Conclusion: Both methods yield identical results, demonstrating that the energy transfer is a function of position, not the route taken.

Summary

The path independence of the work done by the Coulomb force is a cornerstone of electrostatic theory. It not only proves that the electrostatic field is a conservative field but also provides the theoretical justification for the concepts of electric potential energy and electric potential. By converting complex vector-based work calculations into simple scalar potential differences, this principle significantly enhances our ability to analyze and design complex electrical systems.