Question
A point charge of q = +2 µC is placed in free space. Find the electric potential at a point P located r = 0.3 m away from the charge. Also, find the work done in bringing a test charge of q₀ = +1 µC from infinity to point P.
Solution — Step by Step
Electric potential due to a point charge is:
Here, k = 9 × 10⁹ N·m²/C², q is the source charge, and r is the distance from the charge to the point. Unlike electric field, potential is a scalar — no direction needed.
So V = 6 × 10⁴ V at point P.
The relation between work done and potential is:
We use this because potential at infinity is zero, so the potential difference is simply V itself.
W = 0.06 J (or 6 × 10⁻² J).
Both charges are positive. Bringing a positive test charge toward another positive charge requires work done against the repulsive force — so W is positive. The answer checks out physically.
Why This Works
Electric potential at a point is defined as the work done per unit positive charge in bringing a test charge from infinity to that point, with no acceleration (quasi-static process). The formula V = kq/r directly gives us this quantity for a point charge.
The key reason potential varies as 1/r (while electric field varies as 1/r²) is that potential is obtained by integrating the electric field over distance. Each integration step picks up one factor of r in the denominator.
Since potential is a scalar, when multiple charges are present, we simply add the potentials algebraically — no vector addition needed. This is what makes potential extremely useful for problems with complex charge distributions.
For JEE, the scalar nature of potential is a high-weightage concept. Superposition of potentials (just add numbers) versus superposition of electric fields (vector addition) — this distinction has appeared in multiple PYQs.
Alternative Method — Using Energy Directly
Instead of finding V first, we can directly use the definition of potential energy:
Since both charges start at infinity (U = 0) and end with U = 0.06 J, the work done equals this potential energy: W = 0.06 J. Same answer, different route.
This approach is faster when the question directly asks for work done or potential energy between two specific charges.
Common Mistake
Students often confuse electric potential V = kq/r with electric field E = kq/r². The field has r² in the denominator and is a vector. The potential has r (not squared) and is a scalar. In CBSE board exams, writing V = kq/r² is one of the most common errors and costs full marks. Remember: potential comes from integrating the field, which “reduces” the power of r by 1.
A second trap: when the source charge is negative, the potential is negative. A negative potential means we actually get energy back when bringing a positive test charge from infinity — the charges attract. Always carry the sign of q through the calculation.