Electric Potential

Electric potential is one of the most powerful concepts in electrostatics because it lets you describe the 'electrical landscape' of a region of space with a single number at every point. Think of it like altitude on a topographic map: a ball placed at a high altitude naturally rolls downhill toward lower altitude. Similarly, a positive charge placed at a point of high electric potential naturally moves toward regions of lower potential. The electric potential at a point tells you how much electric potential energy each unit of positive charge would have if placed there. Formally, the electric potential V at a point in space is defined as the electric potential energy U of a test charge q divided by the magnitude of that charge: V = U/q. The unit of electric potential is the volt (V), where 1 volt equals 1 joule per coulomb (1 V = 1 J/C). Because potential is energy per charge, it is a scalar quantity — it has magnitude but no direction. This makes it far easier to work with than the electric field, which is a vector. To find the total potential from many charges, you simply add the individual potentials as ordinary numbers, not as vectors. For a single point charge Q, the electric potential at a distance r is given by Coulomb's law in its energy form: V = kQ/r, where k = 8.99 × 10⁹ N·m²/C² is Coulomb's constant. Notice that potential falls off as 1/r rather than 1/r² like the electric field. Also notice the sign matters: a positive charge creates a positive potential around it (a 'hill'), while a negative charge creates a negative potential (a 'valley'). Charges naturally flow from high potential to low potential, and the difference in potential between two points — called the potential difference or voltage — is what drives electric currents in circuits.

Key takeaways
  • Potential Energy vs. Potential
  • Potential Due to a Single Point Charge
  • Superposition: Potential from Multiple Charges
  • Relationship Between Electric Field and Potential

Potential Energy vs. Potential

Electric potential energy U belongs to a system of charges — it's stored in the interaction between them. Electric potential V is a property of a location in space, independent of whether a test charge is actually there. The relationship is U = qV. If a charge q moves through a potential difference ΔV, its change in potential energy is ΔU = qΔV. For example, if a proton (q = +1.6 × 10⁻¹⁹ C) moves through a potential difference of 1 V, it gains or loses 1.6 × 10⁻¹⁹ J of energy — exactly one electron-volt (eV), a unit used constantly in atomic and nuclear physics.

Potential Due to a Single Point Charge

For a point charge Q, the electric potential at distance r is V = kQ/r. This is derived from the work done to bring a test charge from infinity (where V = 0 by convention) to distance r against the Coulomb force. Since the force is conservative, this work is path-independent. For Q > 0, V > 0 everywhere and increases as you get closer. For Q < 0, V < 0 everywhere and becomes more negative closer in. The potential is spherically symmetric — every point on a sphere of radius r around Q has the same potential, forming an equipotential surface.

Superposition: Potential from Multiple Charges

The total electric potential at a point P due to a collection of point charges Q₁, Q₂, Q₃, ... located at distances r₁, r₂, r₃, ... from P is simply the algebraic (scalar) sum: V_total = kQ₁/r₁ + kQ₂/r₂ + kQ₃/r₃ + ... This superposition principle makes potential much easier to calculate than electric field. With electric field, you must add vectors — keeping track of x and y components. With potential, you just add signed numbers. For a system of two equal and opposite charges (an electric dipole), the potential is zero at every point on the perpendicular bisector plane between them, even though the electric field is not zero there.

Relationship Between Electric Field and Potential

The electric field and electric potential are intimately connected. The electric field points in the direction of steepest decrease of potential, just as gravity points downhill. Mathematically, in one dimension, E = −dV/dx. The negative sign is essential: field points from high potential to low potential. Equipotential lines (lines of constant V) are always perpendicular to electric field lines. No work is done when a charge moves along an equipotential surface, because there is no change in potential energy. This is why the surface of a conductor in electrostatic equilibrium is always an equipotential — free charges rearrange themselves until no net work can be extracted by moving along the surface.

The Key Equation

V=kQr=14πε0QrV = \frac{kQ}{r} = \frac{1}{4\pi\varepsilon_0}\frac{Q}{r}
  • V — Electric potential (Volt (V = J/C)): The electric potential energy per unit positive charge at a given point in space.
  • k — Coulomb's constant (N·m²/C²): k = 8.99 × 10⁹ N·m²/C², also written as 1/(4πε₀).
  • Q — Source charge (Coulomb (C)): The charge creating the potential. Positive Q gives positive V; negative Q gives negative V.
  • r — Distance (Meter (m)): The distance from the point charge Q to the point where V is being evaluated.
  • \varepsilon_0 — Permittivity of free space (C²/(N·m²)): ε₀ = 8.85 × 10⁻¹² C²/(N·m²), the fundamental constant describing how electric fields propagate in a vacuum.

Worked Example

Problem. Two point charges are placed on the x-axis: Q₁ = +3.0 μC at x = 0 and Q₂ = −2.0 μC at x = 4.0 cm. (a) Find the electric potential at point P located at x = 8.0 cm. (b) How much work must an external agent do to bring a charge q = +1.0 μC from infinity to point P?

VP=−1.13×105 V,W=−0.113 JV_P = -1.13 \times 10^5 \text{ V}, \quad W = -0.113 \text{ J}

Answer. The electric potential at P is −1.13 × 10⁵ V, and the work done by an external agent bringing +1.0 μC from infinity to P is −0.113 J.

Figure
Diagram
Electric Potential from a Point Charge: Equipotential Surfaces+QV₁ (high)V₂V₃V₄ (low)E fieldlinesDashed circles = equipotential surfaces (V = kQ/r = constant on each circle)Red arrows = E field lines (radially outward, ⊥ to equipotentials)
Origins · 1 minAlessandro Volta and how this was discovered
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Alessandro Volta
1745–1827 · Italian

Volta invented the first electrochemical battery (the voltaic pile) in 1800, creating for the first time a steady source of electric potential difference. His work proved that electricity could be generated chemically and that a sustained potential difference (voltage) could drive a continuous current — laying the foundation for all of electrical engineering. The unit of electric potential, the volt (V), is named in his honor. Napoleon Bonaparte was so impressed by Volta's voltaic pile that he made Volta a count and senator of the Kingdom of Italy. Napoleon personally attended and presided over Volta's demonstration to the French National Institute in 1801.

  • Coulomb's Law — The formula V = kQ/r is derived directly from Coulomb's force law. Understanding the 1/r² force law explains why potential follows a 1/r relationship.
  • Electric Field — Electric potential and electric field are mathematically related by E = −dV/dx. The field points in the direction of steepest decrease of potential.
  • Electric Potential Energy — Electric potential V is defined as potential energy per unit charge (V = U/q). Mastering the distinction between U and V is essential.
  • Voltage and EMF in Circuits — Voltage in circuits is simply potential difference ΔV. The concept of electric potential directly underpins how batteries, resistors, and capacitors behave.