Library / Foundations: Physics / FPH-58
Foundations: Physics
Electric potential of a point charge
Electric potential at a distance r from a point charge Q, taking the potential to be zero infinitely far away.
- Ref
- FPH-58
- Rev
- v1
- Field
- Foundations: Physics
- Result
- V [V]
Q is the source charge in coulombs, signed, and r the distance from it in metres. The potential is zero infinitely far away and has the sign of the charge. The electric field is the slope of this potential: for a point charge E = V/r.
Electric potential at a distance r from a point charge Q, taking the potential to be zero infinitely far away. It is the work per unit charge needed to bring a small positive test charge in from far away.
Calculator
- Electric field at this distance, Q/(4πε₀r²)359.5 V/m
- Energy of a 1 nC test charge placed here1.798×10−7 J
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- At 0.5 m from a 10 nC charge the potential is 179.8 V. Voltages near small charges are large even when the charge itself is tiny.
- Potential falls as 1/r, more slowly than the field (1/r²), so a charge still shapes the voltage far away.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (V) | Local step | Change in result |
|---|---|---|---|---|---|---|
| Q Source charge (signed) | −1e-4 to 1e-4 C | −1,798,000 | 1,798,000 | 3,595,000 | +0.01 C | +1e8 % |
| r Distance from the charge | 0.01 to 10 m | 8.988 | 8,988 | 8,979 | +10% | −9.09 % |
Sweep of Distance from the charge
| Distance from the charge (m) | Electric potential (V) |
|---|---|
| 0.01 | 8,988 |
| 0.02371 | 3,790 |
| 0.05623 | 1,598 |
| 0.1334 | 674 |
| 0.3162 | 284.2 |
| 0.7499 | 119.9 |
| 1.778 | 50.54 |
| 4.217 | 21.31 |
| 10 | 8.988 |
Sweep one variable
Variables and how each one contributes
Electric potential
What the equation gives you. Everything else feeds it.
Source charge (signed)
The source charge in coulombs, with its sign. Raises the potential in direct proportion; a negative charge gives a negative potential. Raising it by 10 % (to 1.1×10−8) moves the result by +10 %, from 179.8 to 197.7 V.
Distance from the charge
The distance from the charge, in m. Lowers the potential in inverse proportion to the distance. Raising it by 10 % (to 0.55) moves the result by −9.09 %, from 179.8 to 163.4 V.
Worked example with the values above
- Constant: 1/(4πε₀) = 8.988 × 10⁹ N·m²/C².
- V = 8.988 × 10⁹ × 1×10−8 / 0.5.
- V = 179.751 V.
Assumptions and limits
- A point charge, or a uniformly charged sphere outside its surface.
- Potential defined as zero at infinity.
- A vacuum or air; in a dielectric replace ε₀ by ε₀εr.
Common mistakes
- Confusing potential V (a scalar, in volts) with the field E (a vector, in V/m).
- Using r from the surface rather than the centre.
- Forgetting that negative charge gives negative potential.
Statistics and uncertainty
The formula is exact for a point charge. A 1 % error in the distance gives a 1 % error in the potential (the potential varies as 1/r, not 1/r²).
Where it is used
- Finding the voltage near a charged conductor or particle.
- Working out the energy needed to move a charge in the field of another.
Maths you need
- Electric field: force per unit charge
- Work and energy: work = charge × potential difference