Calculation Formula for the Electric Potential of a Point Charge

In the study of electromagnetism, electric potential serves as a fundamental scalar quantity that describes the energy state of an electric field at a specific point in space. While the electric field strength tells us about the force exerted on a charge, the electric potential provides a more intuitive "landscape" of the energy available within that field.

To grasp the concept, it is helpful to think of electric potential as the "electrical height" of a point in space. Just as gravitational potential energy depends on an object's height in a gravitational field, electric potential describes the potential energy per unit charge.

The Relationship Between Potential and Potential Energy

It is crucial to distinguish between electric potential ($\phi$ or $V$) and electric potential energy ($E_p$). The potential is a property of the location in the field, whereas the potential energy is a property of a specific charge placed at that location. They are mathematically linked by the following relationship:

$$E_p = q\phi$$

Where:

• $E_p$ is the electric potential energy (Joules, $\text{J}$).
• $q$ is the charge placed at that point (Coulombs, $\text{C}$).
• $\phi$ is the electric potential at that point (Volts, $\text{V}$).

Because the electrostatic force is a conservative force, the work done in moving a charge between two points is independent of the path taken. This property allows us to define a unique scalar value for the potential at every point in an electrostatic field.

Deriving the Formula for a Point Charge

To derive the formula for the potential created by a single point charge, we must establish a reference point. In standard electrostatic calculations, we define the potential at an infinite distance from the charge to be zero: $\phi_{\infty} = 0$.

1. The Role of Work

The electric potential at a distance $r$ from a charge $q$ is defined as the amount of work required to bring a unit positive test charge ($+1\text{C}$) from infinity to that specific point $r$ within the electric field.

2. Mathematical Derivation

We begin with the known expression for the electric field $\vec{E}$ produced by a point charge $q$ at a distance $r$:

$$E = k \frac{q}{r^2}$$

Where $k$ is the electrostatic constant ($k \approx 8.987 \times 10^9 \text{N}\cdot\text{m}^2/\text{C}^2$).

The electric potential $\phi$ is mathematically defined as the negative line integral of the electric field from infinity to the point of interest:

$$\phi = -\int_{\infty}^{r} \vec{E} \cdot d\vec{l}$$

Substituting the expression for $E$ into the integral:

$$\phi = -\int_{\infty}^{r} k \frac{q}{r^2} dr$$

Solving the integral:

$$\phi = -kq \left[ -\frac{1}{r} \right]_{\infty}^{r} = kq \left( \frac{1}{r} - \frac{1}{\infty} \right)$$

Since $1/\infty$ approaches zero, we arrive at the final formula.

3. The Final Calculation Formula

The electric potential $\phi$ at a distance $r$ from a point charge $q$ is:

$$\phi = \frac{1}{4\pi\epsilon_0} \frac{q}{r} = k \frac{q}{r}$$

Variable Definitions:

• $\phi$: Electric potential (Volts, $\text{V}$).
• $k$: Electrostatic constant.
• $\epsilon_0$: Vacuum permittivity ($\approx 8.854 \times 10^{-12} \text{F/m}$).
• $q$: The source charge (Coulombs, $\text{C}$). Note: The sign of $q$ must be included.
• $r$: The radial distance from the charge to the observation point (meters, $\text{m}$).

Critical Characteristics of Electric Potential

When applying this formula in physics problems, three key distinctions are vital for accuracy:

• Scalar Nature: Unlike the electric field $\vec{E}$, which is a vector and requires direction-based addition, the electric potential $\phi$ is a scalar. This significantly simplifies calculations involving multiple charges, as you can use simple algebraic addition.
• The Influence of Charge Sign:
  • For a positive charge ($q > 0$), the potential is positive. The potential decreases as you move further away from the charge.
  • For a negative charge ($q < 0$), the potential is negative. The potential increases (becomes less negative) as you move away, approaching zero from below.
• Rate of Decay: The electric potential follows an inverse relationship with distance ($\phi \propto 1/r$), whereas the electric field follows an inverse-square law ($E \propto 1/r^2$). Consequently, the electric potential decays more slowly across space than the electric field strength.

The Principle of Superposition

In complex systems containing multiple point charges, the total electric potential at any point $P$ is the algebraic sum of the potentials contributed by each individual charge. This is known as the Principle of Superposition.

If a system contains $n$ charges ($q_1, q_2, \dots, q_n$), the total potential $\phi_{total}$ at point $P$ is:

$$\phi_{total} = \sum_{i=1}^{n} k \frac{q_i}{r_i} = k \left( \frac{q_1}{r_1} + \frac{q_2}{r_2} + \dots + \frac{q_n}{r_n} \right)$$

Where $r_i$ represents the distance from the $i$-th charge to point $P$.

Practical Application: A Worked Example

Problem Statement:
Two point charges are placed on the $x$-axis in a vacuum. Charge $q_1 = +2\mu\text{C}$ is at $x = 0$, and charge $q_2 = -2\mu\text{C}$ is at $x = 0.2\text{m}$. Calculate the total electric potential at the midpoint, $x = 0.1\text{m}$.

Step-by-Step Solution:

1. Identify Distances:

   • Distance from $q_1$ to the midpoint ($r_1$): $0.1\text{m}$.
   • Distance from $q_2$ to the midpoint ($r_2$): $0.2\text{m} - 0.1\text{m} = 0.1\text{m}$.

2. Apply the Superposition Formula:
   $$\phi_{total} = k \frac{q_1}{r_1} + k \frac{q_2}{r_2}$$

3. Numerical Calculation:
   $$\phi_{total} = (8.99 \times 10^9) \times \left( \frac{2 \times 10^{-6}}{0.1} + \frac{-2 \times 10^{-6}}{0.1} \right)$$
   $$\phi_{total} = (8.99 \times 10^9) \times (0.02 - 0.02) = 0\text{V}$$

Conclusion: At the exact midpoint between these two equal and opposite charges, the positive and negative potentials cancel each other out perfectly, resulting in a net potential of $0\text{V}$.

Summary Comparison: Electric Field vs. Electric Potential

To avoid common conceptual errors, refer to the following comparison: