define electric potential at a point as the work done per unit positive charge in bringing a small test charge from infinity to the point

Definition

Electric potential at a point is the work done per unit positive charge in bringing a small test charge from infinity to that point. In symbols: \$V = \dfrac{W}{q}\$.

⚡️ Think of a tiny ball (the test charge) that you lift from the ground (infinity) up to a hilltop (the point). The energy you spend lifting the ball is the work done. The potential is that energy divided by the ball’s weight (charge).

Mathematical Expression

For a point charge \$Q\$, the potential at a distance \$r\$ is given by

\$V = \frac{1}{4\pi\varepsilon_0}\,\frac{Q}{r}\$

where \$k = \dfrac{1}{4\pi\varepsilon_0} \approx 8.99\times10^9\;\text{N·m}^2\text{/C}^2\$.

Worked Example

Suppose we have a charge of +5 μC at the origin. What is the potential at a point 0.2 m away?

1. Convert μC to C: \$5\,\mu\text{C} = 5\times10^{-6}\,\text{C}\$.
2. Use the formula: \$V = k\,\dfrac{Q}{r}\$.
3. Insert the numbers: \$V = 8.99\times10^9 \times \dfrac{5\times10^{-6}}{0.2}\$.
4. Calculate: \$V \approx 2.25\times10^5\;\text{V}\$ (225 kV).

🔋 The potential is huge because the charge is small but the distance is tiny.

Exam Tip Box

• Potential is a scalar, not a vector.
• Work done in electrostatics is path‑independent; you can choose any path from infinity.
• Remember the SI units: 1 μC = \$1\times10^{-6}\$ C, 1 V = 1 J/C.
• When given a point charge, always use \$V = kQ/r\$; for a sphere, use \$V = kQ/R\$ outside the sphere.
• Check your arithmetic carefully—large numbers can lead to mistakes.

Potential Table (for +5 μC)

📊 Notice how the potential decreases as you move further from the charge.