Formula for Electric Potential Distribution of a Point Charge

In the study of electrostatics, electric potential ($V$) is a fundamental scalar quantity used to describe the state of a position within an electric field. Unlike the electric field, which is a vector quantity, the potential provides a simpler way to analyze the energy landscape created by charged particles.

By definition, the electric potential at a specific point is the amount of electric potential energy ($U$) per unit charge ($q$) located at that point. The relationship is expressed by the following formula:

[
V = \frac{U}{q}
]

The SI unit for electric potential is the Volt (V), where $1\text{ V} = 1\text{ J/C}$. Conversely, the potential energy of a charge $q$ within a potential $V$ is given by:

[
U = qV
]

While the electric field describes the force exerted on a charge, the electric potential describes the work required to move a charge from a reference point to a specific location.

Deriving the Potential of a Point Charge

To understand how the potential is distributed around a single point charge, we must begin with the known behavior of the electric field.

1. The Coulomb Electric Field

Consider a point charge $Q$ situated at the origin. According to Coulomb's Law, the electric field $\mathbf{E}$ at a distance $r$ in a vacuum is a radial vector field defined as:

[
\mathbf{E}(\mathbf{r}) = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r^{2}}\hat{\mathbf{r}}
]

In this expression:

  • $\varepsilon_0$ is the vacuum permittivity (approximately $8.854 \times 10^{-12} \text{ F/m}$).
  • $r$ is the magnitude of the displacement vector from the charge to the observation point.
  • $\hat{\mathbf{r}}$ is the unit vector pointing radially outward from the charge.

2. The Integral Definition of Potential

Electric potential is mathematically defined as the negative line integral of the electric field from a reference point (usually infinity) to the point of interest:

[
V(\mathbf{r}) = -\int_{\infty}^{\mathbf{r}} \mathbf{E} \cdot d\mathbf{l}
]

Due to the spherical symmetry of a point charge, the most efficient path for this integration is a straight radial line. Substituting the expression for the Coulomb field into the integral, we get:

[
V(r) = -\int_{\infty}^{r} \frac{1}{4\pi\varepsilon_0}\frac{Q}{r'^{2}},dr'
]

3. The Resulting Distribution Formula

Performing the integration yields the formula for the electric potential distribution of a point charge:

[
V(r) = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}
]

This result reveals a critical characteristic: the electric potential of a point charge is inversely proportional to the distance ($1/r$). This is a "slower" decay compared to the electric field, which decays as $1/r^2$.

The Interplay Between Electric Field and Potential

The relationship between $\mathbf{E}$ and $V$ is not merely coincidental; they are two sides of the same physical coin.

  • The Gradient Relationship: The electric field is the negative gradient of the electric potential:
    [
    \mathbf{E} = -\nabla V
    ]
    For a point charge, this simplifies to $\mathbf{E} = -\frac{dV}{dr}\hat{\mathbf{r}}$. If you take the derivative of our derived potential formula, you will mathematically recover the original Coulomb field.

  • Equipotential Surfaces: An equipotential surface is a locus of points where the potential $V$ remains constant. For a single point charge, these surfaces are concentric spheres centered on the charge. Because the potential is the same everywhere on such a sphere, no work is required to move a charge along its surface, meaning the electric field must always be perpendicular to these surfaces.

Practical Applications and Calculations

Example 1: Single Charge Calculation

Suppose we have a point charge $Q = 5,\mu\text{C}$. What is the electric potential at a distance of $10\text{ cm}$ from the charge?

Using the formula:
[
\begin{aligned}
V &= \frac{1}{4\pi\varepsilon_0}\frac{Q}{r} \
&= (9 \times 10^9 \text{ N}\cdot\text{m}^2/\text{C}^2) \frac{5 \times 10^{-6} \text{ C}}{0.10 \text{ m}} \
&= 4.5 \times 10^5 \text{ V}
\end{aligned}
]

Example 2: The Principle of Superposition

In systems with multiple charges, the total potential at any point is the algebraic sum of the potentials created by each individual charge.

Consider two charges: $Q_1 = +2,\mu\text{C}$ at the origin and $Q_2 = -3,\mu\text{C}$ at $x = 0.2\text{ m}$. To find the potential at $x = 0.1\text{ m}$:

  1. Distance from $Q_1$: $r_1 = 0.1\text{ m}$.
  2. Distance from $Q_2$: $r_2 = |0.2 - 0.1| = 0.1\text{ m}$.

The total potential $V_{total}$ is:
[
\begin{aligned}
V &= \frac{1}{4\pi\varepsilon_0}\left(\frac{Q_1}{r_1} + \frac{Q_2}{r_2}\right) \
&= 9 \times 10^9 \left(\frac{2 \times 10^{-6}}{0.1} - \frac{3 \times 10^{-6}}{0.1}\right) \
&= -9 \times 10^4 \text{ V}
\end{aligned}
]
The negative sign indicates that the point is in a region of lower potential relative to infinity, dominated by the negative charge.

Energy Conservation in Electrostatic Systems

When dealing with a collection of point charges, the total electrostatic potential energy ($U_{\text{total}}$) of the system represents the work required to assemble the charges from infinity. It is calculated as the sum of the interaction energies between all pairs:

[
U_{\text{total}} = \frac{1}{4\pi\varepsilon_0}\sum_{i<j}\frac{Q_i Q_j}{r_{ij}}
]

To avoid double-counting the interactions, the total energy can also be expressed using the potential $V_i$ at the location of each charge $i$ (where $V_i$ is the potential created by all other charges):

[
U_{\text{total}} = \frac{1}{2}\sum_i Q_i V_i
]

This formulation is a powerful tool in thermodynamics and circuit analysis, ensuring that energy conservation is strictly maintained within the system.

Critical Nuances and Common Pitfalls

To master electrostatics, one must be aware of several technical subtleties:

  • The Reference Point: While we typically set $V = 0$ at infinity, this is a convention. In specific engineering contexts (like near a conducting plane), a different reference point may be chosen. Consistency is vital.
  • The Singularity Problem (Self-Energy): The formula $V = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$ approaches infinity as $r \to 0$. This implies that a "true" mathematical point charge would have infinite self-energy. In physical reality, charges have a finite size, and we use charge distribution models to resolve this singularity.
  • Medium Effects: The constant $\varepsilon_0$ applies only to a vacuum. If the charges are immersed in a dielectric medium, you must account for the relative permittivity ($\varepsilon_r$), adjusting the formula to $V = \frac{1}{4\pi\varepsilon_r\varepsilon_0}\frac{Q}{r}$.
  • Sign Conventions: Always remember that potential is a scalar. A negative charge produces a negative potential. When using the superposition principle, the signs of the charges must be included in the summation.

Summary

The formula $V(r) = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}$ is a cornerstone of electromagnetism. It describes how the electric potential decays with distance, provides the basis for understanding equipotential surfaces, and allows for the calculation of complex energy states through the principle of superposition. Mastering this distribution is essential for advancing into more complex topics such as capacitance, electromagnetic waves, and field theory.