Electric Potential

Electric potential is a scalar quantity that describes the energy state of a point in an electric field. While the electric field E is a vector that tells us the force per unit charge at a point, the electric potential V tells us the potential energy per unit charge. For complex field configurations, working with potential is far easier because scalar quantities can be added directly without vector decomposition. This makes potential central to circuit theory, capacitor design, and wave propagation.

Core Concept: Potential as Energy per Unit Charge

The electric potential at a point P is defined as the work done by an external agent in bringing a unit positive test charge from a reference point at infinity (where V is defined to be zero) to point P, against the electric force, without any acceleration. Since this work is path-independent in an electrostatic field (because E is a conservative field), V is a well-defined scalar function of position.

A conservative field is one where the work done in moving a charge between two points is independent of the path taken. This property arises because the curl of E is zero for static fields. It is this conservative nature that allows us to define a unique potential at every point in space, and it is what makes circuit laws like Kirchhoff's Voltage Law valid.

The unit of electric potential is the Volt (V), which equals one Joule per Coulomb. When we say a point is at 5V, we mean that 5 joules of work per coulomb are needed to bring a test charge from infinity to that point against the field. A region of high potential has a strong pushing force on positive charges, which will spontaneously move from high to low potential.

Mathematical Expression

The electric potential V at a point is defined by the line integral formula: V equals negative integral of E dot dL from the reference point to the field point. The negative sign arises because work done against the field is positive while the field itself points in the direction of decreasing potential. For a single point charge Q located at the origin, the potential at distance r is V equals Q divided by 4 pi epsilon-naught r.

For a system of multiple point charges, the total potential at any point is the algebraic sum of potentials due to each charge individually. This is the principle of superposition for potential. Because potential is a scalar, this sum is a simple addition of numbers, not vector addition. This is one major practical advantage of working with potential rather than with the electric field directly.

The potential difference between two points A and B is given by V_AB equals V_A minus V_B, which equals the negative line integral of E dot dL from A to B. In circuit analysis, this is the voltage drop from A to B and determines the energy released or absorbed when a charge moves between these points.

Practical Understanding

In electrostatic shielding, a conductor held at a fixed potential (like a grounded metal cage) creates a zero-field interior. All points inside the conductor are at the same potential, and there is no E field to drive current or disturb sensitive equipment. This is the physical basis of the Faraday cage.

In semiconductor physics, the built-in potential across a p-n junction arises from the diffusion of charge carriers creating a charge separation. This potential barrier, typically around 0.7V for silicon, directly determines diode behavior and is calculated using the same potential concepts from electromagnetics.

Given:
Two point charges: Q1 = +4 x 10^-9 C at origin, Q2 = -2 x 10^-9 C at x = 0.3 m.
Find the electric potential at point P located at x = 0.1 m.

Why this formula applies:
Superposition for scalar potential. V_total = V1 + V2.
V due to each charge = Q / (4*pi*epsilon_0*r).

Formula:
V = (1/4*pi*epsilon_0) * [Q1/r1 + Q2/r2]

Substitution:
r1 = 0.1 m (distance from Q1 to P)
r2 = 0.3 - 0.1 = 0.2 m (distance from Q2 to P)
k = 9 x 10^9 N.m^2/C^2

Calculation:
V1 = 9e9 * 4e-9 / 0.1 = 360 V
V2 = 9e9 * (-2e-9) / 0.2 = -90 V
V_total = 360 + (-90) = 270 V

Final Answer: V at P = 270 V

Exam Tip: For GATE problems involving potential, use V = kQ/r and scalar superposition. Do not add potentials as vectors. If work done is asked: W = q * (V_final - V_initial). A positive charge moves spontaneously from high V to low V (losing potential energy).

• V is a scalar: it has no direction, only magnitude and sign. Positive near positive charges, negative near negative charges.
• Equipotential surfaces are surfaces of constant V. No work is done in moving a charge along an equipotential surface.
• E field lines are always perpendicular to equipotential surfaces, pointing from high V to low V.
• For a conductor at equilibrium, the entire surface and interior are at the same potential (equipotential body).
• Work done in moving charge q from V_A to V_B: W = q(V_A - V_B). Positive work means charge moves from high to low potential.

Quick Revision

• V = -integral(E dot dL) from reference to point. Reference is usually at infinity where V = 0.
• For point charge: V = Q / (4*pi*epsilon_0*r) = kQ/r.
• Superposition: V_total = V1 + V2 + V3 ... (scalar sum, no vector addition needed).
• Work: W = q * delta_V. Positive charges move from high V to low V spontaneously.
• E is perpendicular to equipotential surfaces. No work along equipotential.
• Exam trap: Potential can be zero where E is non-zero (e.g., midpoint between equal and opposite charges has V=0 but E is non-zero).