Electric Potential Due to a System of Charges

Physics · Electrostatic Potential And Capacitance · NEET

The electric potential at a point due to a system of charges is just the ALGEBRAIC sum of the potentials from each charge alone: V = (1/4πε₀)(q₁/r₁ + q₂/r₂ + ... + qₙ/rₙ). This is the superposition principle. Memory hook: "Potential is a scalar, so you ADD numbers with their + or − signs, no arrows, no angles."
Potential at P = k(q₁/r₁ + q₂/r₂ + q₃/r₃)P+q₁+q₂−q₃r₁r₂r₃Add signed scalars — no arrows, r is always positive
Each charge sends a signed potential term kq/r to point P; the total potential is their algebraic sum (superposition). Note −q₃ contributes a negative term.

Your doubts, answered

Do I add potentials as vectors like I do for electric field?

No. Electric potential is a SCALAR. For a system of charges you add the individual potentials as plain numbers, keeping each charge's + or − sign: V = kq₁/r₁ + kq₂/r₂ + ... There are no components, no arrows and no angles between them. This is why potential problems are usually easier than field problems for the same charge arrangement.

How do I put the sign of a negative charge into the formula?

Use the charge with its own sign inside V = kq/r. For a −2 μC charge you write V = k×(−2×10⁻⁶)/r, which gives a negative potential. Then add it to the other terms. The distance r is always taken as a positive number (a magnitude); only the charge carries the sign.

What is the potential at the exact centre of a square with charges at the corners?

Every corner charge is the same distance from the centre, r = (side)/√2 (half the diagonal). So V_centre = k(q₁+q₂+q₃+q₄)/r. Just add the corner charges with signs, then divide by that one distance. If the four charges add to zero, the potential at the centre is zero even though the field there may not be.

The field is zero at a point but the potential is not zero. Is that possible?

Yes, and NEET loves this. Field E is a vector, potential V is a scalar. Two equal positive charges: at the midpoint the two field vectors cancel (E = 0) but the two potentials ADD (V ≠ 0). Field zero does not force potential zero, and potential zero does not force field zero. Treat them separately.

Which distance do I use, straight-line or along a path?

Always the straight-line (shortest) distance from each charge to the point where you want the potential. Potential depends only on that direct distance r, not on any path you imagine walking. This follows from the electrostatic force being conservative.

⚠️ The NEET trap
Adding the potentials as vectors, or ignoring the negative sign of a charge and taking every term as positive.
Potential is a scalar: add kq/r terms as signed numbers (negative charge gives a negative term). Distance r is always positive.
🧠 Field = vectors (arrows add). Potential = scalars (numbers add). If you drew an arrow for potential, you already made the mistake.

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Frequently asked

What is the formula for electric potential due to a system of charges?

V = (1/4πε₀) Σ qᵢ/rᵢ = k(q₁/r₁ + q₂/r₂ + ... + qₙ/rₙ), where k = 9×10⁹ N·m²/C², each qᵢ is taken with its own sign, and rᵢ is the straight-line distance of charge qᵢ from the point.

Why can we simply add the potentials?

Because of the superposition principle and because potential is a scalar built from conservative forces. The work done per unit charge from each source adds independently, so the total potential is the algebraic sum of the individual potentials.

Is the total potential ever zero?

Yes. If the signed terms cancel, for example equal + and − charges equally distant from the point, the net potential is zero at that point. This is common at the midpoint of a dipole's perpendicular bisector or the centre of a symmetric charge set that sums to zero.

Does a system of charges have direction for potential?

No. Potential has magnitude and sign only, no direction. Only the electric field of the system has direction. This is the single most important idea to separate the two in NEET problems.

What are the SI units of the potential of a system of charges?

The same as any potential: volt (V), where 1 V = 1 joule per coulomb (J/C). Each kq/r term already comes out in volts, and their sum is in volts.