Electric Potential — Core Principles

Core Principles

Electric potential ($V$) is a scalar quantity representing the work done per unit positive test charge to bring it from infinity to a specific point in an electric field, without acceleration. Its unit is the Volt (V), where $1\,\text{V} = 1\,\text{J/C}$.

The potential due to a point charge $Q$ at a distance $r$ is $V = \frac{1}{4\pi\epsilon_0} \frac{Q}{r}$. For a system of charges, the total potential is the algebraic sum of individual potentials. Electric potential difference ($\Delta V$) between two points A and B is the work done by an external agent per unit charge to move it from A to B.

Equipotential surfaces are regions where the potential is constant, and electric field lines are always perpendicular to them. The electric field $\vec{E}$ is the negative gradient of the potential, $\vec{E} = -\nabla V$.

Electric potential energy ($U$) of a charge $q$ at a potential $V$ is $U = qV$. For a system of two charges $q_1, q_2$ separated by $r$, $U = \frac{1}{4\pi\epsilon_0} \frac{q_1 q_2}{r}$. A dipole $\vec{p}$ in an external field $\vec{E}$ has potential energy $U = -\vec{p} \cdot \vec{E}$.

Important Differences

vs Electric Potential Energy

Aspect	This Topic	Electric Potential Energy
Definition	Electric Potential (V): Work done per unit positive test charge to bring it from infinity to a point.	Electric Potential Energy (U): Work done to bring a specific charge from infinity to a point in an electric field, or to assemble a system of charges.
Unit	Volt (V) or Joules per Coulomb (J/C)	Joule (J)
Nature	Scalar quantity	Scalar quantity
Dependency	Property of the electric field at a point, independent of the charge placed there.	Depends on the magnitude of the charge and the electric potential at its location ($U=qV$). Also depends on the configuration of charges in a system.
Analogy	Gravitational potential (height) at a point.	Gravitational potential energy (mgh) of an object at that height.

Electric potential is a fundamental characteristic of an electric field at a specific location, representing the potential energy that a unit positive charge would possess if placed there. It's a 'per unit charge' concept. Electric potential energy, on the other hand, is the actual energy stored within a specific charge or a system of charges due to their positions within an electric field. While both are scalar quantities, potential is independent of the test charge, whereas potential energy directly depends on the magnitude of the charge involved. Understanding this distinction is crucial for solving problems in electrostatics.