medium difficulty9 min

Potential Due to a System of Charges and an Electric Dipole

The potential at a point due to a system of charges is the algebraic sum of potentials due to individual charges. For an electric dipole, potential depends on distance from the dipole and the angle with its axis.

In one line: For potential, signs decide addition; for dipole, cos θ decides whether the potential survives or becomes zero.

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Understand it

Superposition is simpler for potential than for field because potential is scalar.
For charges q1, q2, q3 at distances r1, r2, r3 from a point, V = k(q1/r1 + q2/r2 + q3/r3).
For a short dipole, V = k p cos θ/r², where p is dipole moment and θ is the angle from the dipole axis.
On the axial line, potential is non-zero; on the equatorial line, the potentials due to +q and -q cancel.

Example

At a point on the perpendicular bisector of a dipole, the distances from +q and -q are equal. Their potentials are equal in magnitude and opposite in sign, so the net potential is zero.

Formula and rule

For a system, V = sum Vi = (1/(4 π ε₀)) sum(qi/ri), with V in volt, qi in coulomb, and ri in metre.

For a short dipole, V = (1/(4 π ε₀)) p cos θ/r², where p = q(2a) in coulomb metre.

This approximation is used when r is much larger than the dipole separation.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Two charges +3 microcoulomb and -1 microcoulomb are 0.20 m and 0.10 m from point P respectively.

1V = 9.0 × 10⁹[(3 × 10⁻⁶)/0.20 + (-1 × 10⁻⁶)/0.10] = 9.0 × 10⁹(15 × 10⁻⁶ - 10 × 10⁻⁶) = 4.5 × 10⁴ V.
2The net potential is positive.

Method you can reuse

List each charge and its distance from the point; write each kq/r term with sign; add algebraically; for dipole questions, identify θ; use cos θ; apply axial or equatorial simplification only when the geometry matches.

Visual tip

For a dipole diagram, mark -q, +q, dipole length 2a, dipole moment p from -q to +q, axial line, equatorial line, point P, distance r, and angle θ.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is the potential zero at a point on the equatorial line of an electric dipole?

Think first, then show answer

The point is equally distant from +q and -q, so the potentials kq/r and -kq/r have equal magnitude and opposite signs. Their algebraic sum is zero.

Common wrong answer

Adding magnitudes only gives a wrong positive potential for systems containing negative charges.

Don't mix this up

A common error is to say the electric field is also zero on the equatorial line because potential is zero. Potential can be zero while electric field is not zero.

Exam tip

For dipole potential, check whether the point is axial, equatorial, or at a general angle before applying the formula.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

It is asked through superposition numericals, axial and equatorial dipole comparisons, assertion-reason questions on zero potential, and short derivation-based questions for dipole potential.

1
Enough for 1 mark
The potential at a point due to a system of charges is the algebraic sum of potentials due to individual charges. For an electric dipole, potential depends on distance from the dipole and the angle with its axis.
2
Add for 2 marks
Example: At a point on the perpendicular bisector of a dipole, the distances from +q and -q are equal. Their potentials are equal in magnitude and opposite in sign, so the net potential is zero.
3
Add for 3 marks
Superposition is simpler for potential than for field because potential is scalar.

Practice

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