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Electric Potential (Definition)

Electric potential at point P is the work done per unit positive charge in bringing a test charge from infinity to P against the electric field. SI unit: volt (V = J C⁻¹).
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Derivation

Motivation

The electric field E⃗\vec{E} tells us the force per unit charge at every point. We want a scalar quantity that encodes the same information more conveniently — one that makes energy calculations straightforward.

Definition via work

The work done by the external agent in moving a unit positive test charge from infinity to point P, without acceleration, is defined as the electric potential at P:

V(P)=W∞→Pq0V(P) = \frac{W_{\infty \to P}}{q_0}

Since the field does work WfieldW_{field} and the external agent does Wext=−WfieldW_{ext} = -W_{field}, and using the work-energy theorem with zero kinetic energy change:

V(P)=−∫∞PE⃗⋅dr⃗V(P) = -\int_{\infty}^{P} \vec{E} \cdot d\vec{r} V=−∫∞PE⃗⋅dr⃗\boxed{V = -\int_{\infty}^{P} \vec{E} \cdot d\vec{r}}

Why scalar

Potential is a scalar — it has magnitude but no direction. Superposition of potentials is algebraic, not vectorial. This makes it far easier to handle multi-charge configurations than working with fields directly.

Units

[V]=JC=V (volt)[V] = \frac{\text{J}}{\text{C}} = \text{V (volt)}
Remember
Potential is defined relative to a reference point. In electrostatics, infinity is the universal reference ($V = 0$ at $\infty$). In circuits, the ground node is taken as reference.