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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.

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Student-friendly explanation

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 theta/r^2, where p is dipole moment and theta 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.

How to write this in exams

  1. 1

    Start with the exact idea

    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. 2

    Then show how to use it

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

  3. 3

    Add one concrete 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.

  4. 4

    Avoid this incomplete answer

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

Definition

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.

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.

Rule to remember

For a system, V = sum Vi = (1/(4 pi epsilon0)) sum(qi/ri), with V in volt, qi in coulomb, and ri in metre. For a short dipole, V = (1/(4 pi epsilon0)) p cos theta/r^2, where p = q(2a) in coulomb metre. This approximation is used when r is much larger than the dipole separation.

Memory hook

For potential, signs decide addition; for dipole, cos theta decides whether the potential survives or becomes zero.

Examples and method

Worked example

Two charges +3 microcoulomb and -1 microcoulomb are 0.20 m and 0.10 m from point P respectively. V = 9.0 x 10^9[(3 x 10^-6)/0.20 + (-1 x 10^-6)/0.10] = 9.0 x 10^9(15 x 10^-6 - 10 x 10^-6) = 4.5 x 10^4 V. The net potential is positive.

Method to apply

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

Diagram support

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 theta.

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.

Avoid common mistakes

Common confusion

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.

Common wrong answer

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

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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-mark answer

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. For a system, V = sum Vi = (1/(4 pi epsilon0)) sum(qi/ri), with V in volt, qi in coulomb, and ri in metre. For a short dipole, V = (1/(4 pi epsilon0)) p cos theta/r^2, where p = q(2a) in coulomb metre. This approximation is used when r is much larger than the dipole separation. 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-mark answer

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 theta/r^2, where p is dipole moment and theta 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. For a system, V = sum Vi = (1/(4 pi epsilon0)) sum(qi/ri), with V in volt, qi in coulomb, and ri in metre. For a short dipole, V = (1/(4 pi epsilon0)) p cos theta/r^2, where p = q(2a) in coulomb metre. This approximation is used when r is much larger than the dipole separation. Two charges +3 microcoulomb and -1 microcoulomb are 0.20 m and 0.10 m from point P respectively. V = 9.0 x 10^9[(3 x 10^-6)/0.20 + (-1 x 10^-6)/0.10] = 9.0 x 10^9(15 x 10^-6 - 10 x 10^-6) = 4.5 x 10^4 V. The net potential is positive. 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. Adding magnitudes only gives a wrong positive potential for systems containing negative charges.
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