The electric potential of a point charge is the scalar value representing the work done per unit charge to move a small positive test charge from infinity to a specific distance away from the source charge. It depends only on the magnitude of the source charge and the distance from it, not on the path taken.
This quantity is measured in volts (joules per coulomb) and describes how much potential energy a unit charge would have at a given location in the electric field generated by a single charged particle. Unlike the electric field, which is a vector, electric potential is a scalar-making it easier to combine across multiple charges.
For a point charge Q, the electric potential V at a distance r is given by the formula:
V = kQ / rwhere k is Coulomb's constant (8.988 × 10⁹ N·m²/C²), Q is the charge in coulombs, and r is the distance in meters from the charge. This equation applies only in vacuum or air; in materials, permittivity alters the effective value of k.
Electric potential arises because a point charge creates an electric field that exerts force on other charges. The potential measures the energy available to do work on a test charge placed in that field. As the test charge moves closer to the source, the potential increases if the source is positive and decreases if negative.
Work is done against the field when bringing a positive test charge toward a positive source charge. Since work is required, the potential is positive. Conversely, moving a positive test charge toward a negative source charge releases energy-the potential becomes negative.
Potential is defined relative to infinity, where it is set to zero. This reference point simplifies calculations and ensures consistency across different systems. The potential at any finite distance is always relative to this baseline.
There is only one fundamental form of electric potential for a point charge: the inverse-distance dependence described by V = kQ/r. However, the sign and magnitude depend on the charge's polarity and size.
| Charge Type | Sign of Potential | Behavior with Distance | Physical Interpretation |
|---|---|---|---|
| Positive (e.g., proton) | Positive | Decreases as 1/r | Repels positive test charges; potential energy rises as they approach |
| Negative (e.g., electron) | Negative | Increases toward zero as r grows | Attracts positive test charges; potential energy drops as they approach |
For multiple point charges, potentials add algebraically (not vectorially), making superposition simpler than with electric fields. For example, two equal positive charges separated by 1 meter create a potential of 2×(kQ/0.5) at their midpoint-twice the potential of either alone.
Electric potential of a point charge underpins many areas of physics and engineering. In atomic models, the potential around a nucleus determines electron energy levels. In quantum mechanics, solving the Schrödinger equation for hydrogen relies on the 1/r potential from the proton.
In circuit design, potential differences drive current flow. Even though circuits involve distributed charges, the concept begins with point-like sources. The voltage across a capacitor, for instance, relates directly to the potential difference created by separated charges.
A practical example: a single electron (charge −1.602 × 10⁻¹⁹ C) generates a potential of −1.44 volts at a distance of 1 nanometer. This value is critical in nanoscale electronics, where tiny charge distributions influence transistor behavior. At 10 nm, the potential drops to −0.144 V-still measurable with sensitive instruments.
Many confuse electric potential with electric field strength. While both stem from the same charge, the field is a vector (direction matters), and potential is a scalar (magnitude only). A region can have zero field but nonzero potential-such as at the center of a uniformly charged spherical shell.
Another error: assuming potential equals force. Potential is energy per charge, not force. Force is related to the gradient (rate of change) of potential. The electric field E is the negative derivative of V: E = −∇V. So high potential doesn't imply strong force-only a steep change in potential does.
Also, potential is not stored energy-it's potential energy per unit charge. Total potential energy U of a test charge q at position r is U = qV. Confusing V with U leads to incorrect units and flawed reasoning.
Consider the hydrogen atom: a proton (charge +1.602 × 10⁻¹⁹ C) at the nucleus and an electron orbiting at about 5.3 × 10⁻¹¹ m (Bohr radius). The electric potential due to the proton at that distance is:
V = (8.988 × 10⁹) × (1.602 × 10⁻¹⁹) / (5.3 × 10⁻¹¹)
≈ 27.2 voltsThis 27.2 V potential explains why electrons in ground state hydrogen have binding energy of approximately 13.6 eV-the energy needed to remove the electron. The potential energy of the electron is qV = (−1.602 × 10⁻¹⁹ C) × 27.2 V ≈ −4.36 × 10⁻¹⁸ J, or −27.2 eV. Half of this is kinetic energy, consistent with the virial theorem.
I've used this calculation in lab simulations of electron trajectories in ion traps. The accuracy of predicted motion hinges on correctly modeling the 1/r potential, especially when simulating collisions between ions and electrons.
While both are derived from Coulomb's law, the electric field E points radially outward from a positive charge and falls off as 1/r². Electric potential V falls off as 1/r. This slower decay means potential extends farther into space than field strength.
For example, at 10 meters from a 1 μC charge, E ≈ 898.8 N/C, while V ≈ 8988 V. The potential remains significant even when the field is weak. This is why high-voltage power lines can induce currents in nearby conductors-even at distances where the field is barely detectable.
Potential is continuous across boundaries (like between air and glass), whereas the electric field can jump discontinuously due to surface charges. This continuity makes potential ideal for numerical methods like finite element analysis in electromagnetics.
According to Griffiths' *Introduction to Electrodynamics* (4th ed.), "The electric potential is defined such that the field is the negative gradient of the potential." This formalism ensures conservation of energy in electrostatic systems. The book emphasizes that potential is a more convenient tool than field for solving boundary-value problems.
The National Institute of Standards and Technology (NIST) confirms that Coulomb's constant k is derived from the permittivity of free space ε₀: k = 1/(4πε₀). NIST values list ε₀ = 8.854 × 10⁻¹² F/m, yielding k = 8.98755 × 10⁹ N·m²/C²-used in precision measurements.
When calculating potential, always verify units: charge in coulombs, distance in meters, and result in volts. Mixing nanometers with coulombs without conversion causes errors by a factor of 10⁹.
The electric potential of a point charge is foundational to understanding how charges interact without direct contact. It enables engineers to model battery voltages, physicists to predict atomic structure, and biologists to study ion transport across cell membranes. Its simplicity belies its power across disciplines.