Login Free Test Start Mock

Physics · Electrostatic Potential and Capacitance

Electrostatic Potential

Electrostatic potential is the energy bookkeeping tool of the electric field: the work done by an external agent in carrying a unit positive charge, without acceleration, from infinity to a given point. NCERT Section 2.2 builds it directly from the conservative nature of the Coulomb force, and NIOS Section 16.1 frames it as the analogue of water level or temperature that decides which way charge tends to flow. For NEET it anchors every later result in the chapter — potential due to a point charge, dipoles, equipotential surfaces and capacitors all descend from this one scalar.

From potential energy to potential

The Coulomb force between two stationary charges is a conservative force. It shares its inverse-square character with gravity, differing only in the proportionality constant, with charges in place of masses. Just as a mass in a gravitational field has gravitational potential energy, a charge in an electrostatic field possesses electrostatic potential energy. This is the starting point of NCERT Section 2.1.

Imagine a fixed charge $Q$ at the origin and a small test charge $q$ that we carry from a point R to a point P against the electric force on it. We apply an external force just equal and opposite to the electric force, $\mathbf{F}_{ext} = -\mathbf{F}_E$, so there is no net force and the charge moves with infinitesimally slow, constant speed. The work done by this external force is stored as potential energy. Defining the potential energy difference,

$$\Delta U = U_P - U_R = W_{RP}$$

where $W_{RP}$ is the work done by the external force in moving $q$ from R to P. Because the right-hand side depends only on the end points and not on the route taken, the potential energy is a meaningful, single-valued quantity.

Figure 1 · Moving a charge does work +Q fixed at origin R P test charge +q F_ext (we push) F_E (repulsion) W_RP done against F_E is stored as ΔU = U_P − U_R

Defining electrostatic potential

The work done on the test charge is proportional to $q$, since the force at any point is $q\mathbf{E}$. It is therefore convenient to divide the work by $q$, so the resulting quantity becomes a property of the field alone, independent of the test charge used to probe it. This ratio is the electrostatic potential $V$. From the work expression NCERT writes

$$V_P - V_R = \frac{U_P - U_R}{q} = \frac{W_{RP}}{q}$$

Choosing the potential to be zero at infinity and bringing the charge from infinity to a point P gives the clean operational definition: the electrostatic potential at any point in a region of electrostatic field is the work done in bringing a unit positive charge, without acceleration, from infinity to that point.

Definition. $V = \dfrac{W}{q_0}$ — the electrostatic potential at a point equals the work done by an external force per unit positive charge in carrying that charge, without acceleration, from infinity to the point. SI unit: volt (V).

NIOS Section 16.1 reaches the same statement and adds the sign convention: the potential at a point is positive when work is done against the field by a positive charge, and negative when the field itself does the work in bringing the unit positive charge from infinity to the point.

Potential difference and the volt

For two points A and B in a field, if an external force carries a test charge $q_0$ from A to B, NIOS writes the work as $W_{AB} = q_0\,(V_B - V_A)$. Rearranging gives the potential difference,

$$V_{AB} = V_B - V_A = \frac{W_{AB}}{q_0}$$

The SI unit of both potential and potential difference is the volt, named after Alessandro Volta. One volt is one joule per coulomb.

QuantitySymbolDefining relationSI unitNature
Electrostatic potentialVV = W/q0 (from infinity)volt (V) = J/Cscalar
Potential differenceV_B − V_AW_AB/q0volt (V)scalar
Potential energyUU = qVjoule (J)scalar
Work doneW_ABq0 (V_B − V_A)joule (J)scalar

One volt of potential difference therefore has a precise meaning: if one joule of work is done in taking one coulomb of charge from one point to another in the field, the points differ in potential by one volt. If that one joule moves the coulomb from infinity to a point, the potential at that point is one volt.

NEET Trap

Potential is a scalar — never add it like a vector

Because $V$ is defined as work divided by charge, both scalars, it has magnitude and sign but no direction. When several charges are present, their potentials add as ordinary signed numbers. Students often try to resolve potentials into components or apply the parallelogram rule, which is wrong.

Add potentials algebraically: $V = V_1 + V_2 + V_3 + \dots$, keeping each sign. No vectors, no components.

Build on this

Once the definition is clear, the next step is to evaluate it for a single charge — see Potential Due to a Point Charge.

A scalar from a conservative field

The whole construction hangs on one property: the electrostatic force is conservative. The work done by the field in moving a charge between two points depends only on the initial and final positions, not on the path. This is precisely what allows a single-valued potential difference to be assigned to a pair of points; if the work depended on the path, the concept of potential would not be meaningful.

This path-independence is also why NIOS calls the electric field a conservative field. It is the same logic that lets us define gravitational potential, transplanted to electrostatics through the shared inverse-square form of the two forces.

Figure 2 · Potential falls with distance V r V → 0 as r → ∞ (reference) large V close to a positive charge

The reference at infinity and U = qV

Potential energy, and hence potential, is fixed only up to an additive constant — adding the same constant at every point leaves all differences unchanged, so the absolute value carries no physical meaning by itself. What is physically significant is the difference. To pin down an absolute value we choose a reference, and the convenient choice is to set the potential to zero at infinity. With $V(\infty)=0$, the potential at a point becomes simply the work to bring a unit positive charge from infinity to that point.

The link back to energy is direct. Multiplying the potential at a point by the charge placed there gives the potential energy of that charge: $U = qV$. Potential is the per-unit-charge property of the location; potential energy is the energy of a particular charge sitting at that location.

NEET Trap

Potential and potential energy are not the same thing

$V$ exists at a point even if no charge is placed there, and is measured in volts. $U = qV$ is the energy of a charge $q$ actually present, measured in joules. A point can have a high potential while the potential energy of a charge there is small if the charge is small. Watch the units to tell them apart.

$V$ in volts (J/C) — property of the point. $U = qV$ in joules — energy of the charge at the point.

NEET Trap

Mind the sign of the work

The defining work is done by the external force, equal and opposite to the electric force, with the charge moved without acceleration. Carrying a positive charge to a region of higher potential needs positive external work; if the field does the moving, the external work is negative. Confusing the work done by the field with the work done by the external agent flips the sign of $\Delta U$.

$W_{ext} = q_0(V_B - V_A) = -W_{field}$ for motion without acceleration.

Quick Recap

Electrostatic Potential in one screen

  • Electrostatic potential $V$ = work done by an external force per unit positive charge to bring it, without acceleration, from infinity to a point: $V = W/q_0$.
  • Potential difference: $V_B - V_A = W_{AB}/q_0$; SI unit volt, $1\text{ V} = 1\text{ J/C}$.
  • $V$ is a scalar — add potentials of several charges algebraically, with signs; never as vectors.
  • The Coulomb force is conservative, so work is path-independent; this is what makes a single-valued potential possible.
  • Reference is $V(\infty)=0$; only potential differences are physically significant.
  • Energy link: $U = qV$ — potential is per unit charge, potential energy belongs to a specific charge.

NEET PYQ Snapshot — Electrostatic Potential

Questions that test the definition, the scalar nature and the path-independence of potential.

NEET 2017

The diagrams show regions of equipotentials at 10 V, 20 V, 30 V and 40 V. A positive charge is moved from A to B in each diagram. Which statement is correct?

  • (1) Maximum work is required to move q in figure (b).
  • (2) Maximum work is required to move q in figure (c).
  • (3) In all four cases the work done is the same.
  • (4) Minimum work is required to move q in figure (a).
Answer: (3)

Work depends only on end points: $W = q(V_B - V_A)$. In every diagram A and B sit on the same pair of equipotentials, so $V_B - V_A$ is identical and the work is the same. Path and arrangement do not matter — the field is conservative.

NEET 2024

A thin spherical shell is charged by some source. The potential difference between the centre C and a point P on the surface is (R = 3 cm, q = 1 μC, $1/4\pi\varepsilon_0 = 9\times10^9$ SI units):

  • (1) $3\times10^5$ V
  • (2) $1\times10^5$ V
  • (3) $0.5\times10^5$ V
  • (4) Zero
Answer: (4)

Inside a charged conducting shell the field is zero, so no work is needed to move a unit charge from the surface to the centre. With $V_C - V_P = -\int \mathbf{E}\cdot d\mathbf{r} = 0$, the potential difference is zero — the interior is at the same potential as the surface.

FAQs — Electrostatic Potential

Definition, scalar nature, the reference at infinity and the energy link, distilled.

What is electrostatic potential at a point?
Electrostatic potential V at a point is the work done by an external force in bringing a unit positive charge, without acceleration, from infinity to that point. Equivalently, V = W/q0, where W is the work to move charge q0 from infinity to the point. Its SI unit is the volt (1 V = 1 J/C).
Is electrostatic potential a scalar or a vector?
Electrostatic potential is a scalar quantity. It is defined through work done divided by charge, both of which are scalars, so it has no direction. Potentials due to several charges add algebraically as ordinary numbers, with their own signs, and never by vector addition.
What is the difference between electric potential and electric potential energy?
Electric potential V is a property of a point in the field alone, measured per unit charge, with SI unit volt (J/C). Electric potential energy U is the energy of a specific charge q placed at that point, given by U = qV, with SI unit joule. Potential exists even where no charge is placed; potential energy needs a charge to be present.
Why is the electrostatic potential at infinity taken as zero?
Only potential differences are physically significant; the absolute value is fixed only after choosing a reference. Infinity is the natural and convenient reference because the Coulomb field of any bounded charge configuration vanishes there, so taking V(infinity) = 0 makes the potential at a point equal to the work done in bringing a unit positive charge from infinity to that point.
Why is work done in moving a charge between two points independent of the path?
The Coulomb force is conservative, like gravity, because of its inverse-square nature. For a conservative force the work done depends only on the initial and final positions, not on the route taken. This path-independence is exactly what lets us define a single-valued potential and potential difference between two points.
What does one volt of potential difference mean?
The potential difference between two points is one volt if one joule of work is done by an external force in moving one coulomb of charge from one point to the other without acceleration. In symbols, V_B − V_A = W_AB / q0, so 1 V = 1 J/C.
Related Topics
Electrostatic Potential and Capacitance Back to the full chapter overview and all subtopics. Potential Due to a Point Charge Evaluate V = kq/r from the definition. Potential Due to a Dipole Superpose two point-charge potentials. Equipotential Surfaces Surfaces of constant V, perpendicular to E. Potential Energy of a Charge System From V to the energy U = qV of assemblies. Electrostatics of Conductors Constant potential inside a conductor.