Potential Due to Point Charges and Dipoles Simple Notes

The potential due to a system of point charges is the algebraic sum of potentials from each charge, because potential is a scalar. For an electric dipole, potential has a special angular dependence and falls faster with distance than a single charge. These formulas let us find voltage anywhere without vector addition difficulty.

This is useful because most real charge systems are collections of points or can be approximated as dipoles. Molecules, antennas, and atomic systems are modeled as dipoles. Knowing their potential helps predict interaction energy, field patterns, and how materials respond in dielectrics.

Key Concepts

Point charge potential: V = k q / r, positive if q positive, negative if q negative. Zero reference at infinity.

Superposition for potential: V_total = Σ k qi / ri. No components needed, just signs and distances to the point.

Electric dipole: Pair +q and -q separated by 2a, dipole moment p = q(2a) from - to +.

Dipole potential at a point distance r from centre, angle theta between r and p: depends on cos theta. Positive on the side of +q, negative on -q side, zero on equatorial plane.

Field can be derived from potential: E components are negative gradients of V. Dipole field falls as 1/r^3, potential as 1/r^2.

Formulas

System of point charges: V(P) = Σ (1/(4 pi epsilon0)) qi / ri. qi in C, ri in m, V in volts.

Dipole potential (r >> a): V = (1/(4 pi epsilon0)) (p cos theta) / r^2 = k p cos theta / r^2. Theta measured from dipole axis (p direction). Maximum on axis (theta 0), zero at equator (theta 90 degrees).

On axis special: theta = 0 => V_axis = k p / r^2. On equatorial line theta = 90 => V = 0 for ideal dipole.

Assumptions: points are in vacuum/air, r large compared to dipole separation for approximate formula. Exact dipole potential uses distances to each charge: V = kq (1/r_+ - 1/r_-).

Worked Example

Charges q1 = +3 microC at distance 0.30 m and q2 = -3 microC at distance 0.60 m from point P. Find total potential at P. Also compute dipole approx if they form dipole with p = 6 × 10^-8 C m, r = 0.50 m, theta = 0.

Step 1: V1 = k q1 / r1 = 9e9 × 3e-6 / 0.30 = 27000 / 0.30? Actually 9e9*3e-6 = 27000. Divide 0.30 => 90,000 V.

Step 2: V2 = k q2 / r2 = 9e9 × (-3e-6) / 0.60 = -27000 / 0.60 = -45,000 V.

Step 3: V_total = 90,000 - 45,000 = 45,000 V (scalar sum). If distances were equal, they'd cancel to zero.

Step 4: Dipole case: V = k p cos theta / r^2 = 9e9 × 6e-8 × 1 / 0.25 = 540 / 0.25 = 2160 V. At equator theta 90, V = 0, though field is non-zero there.

Common Misconceptions

• Misconception: Potential due to dipole zero everywhere on equator means field also zero there. Correction: Field is not zero on equatorial line; V = 0 is a value, field depends on change of V. E_eq = k p / r^3 non-zero.
• Misconception: We need vector sum for potentials like fields. Correction: Potentials add algebraically with sign. This is the main advantage of potential method.
• Misconception: Dipole potential falls as 1/r like point charge. Correction: Because + and - contributions nearly cancel at large r, leading term falls as 1/r^2, field as 1/r^3, faster than monopole.

Real-World Applications

Molecular interactions: Polar molecules modeled as dipoles; potential predicts hydrogen bonding and solubility.

Heart ECG modeling: Heart activity approximated as changing dipole, potential on skin measured.

Antenna theory: Short dipole potential/field formulas design radio transmitters.

Dielectric materials: External field interacts with molecular dipole potentials to produce polarization.

Mass spectrometry and electron optics: Potentials of charge collections shape particle trajectories.

Add point potentials as simple numbers, and a dipole's potential reveals direction: positive lobe, negative lobe, zero plane between. That scalar map then gives field by its slope.