Playground
Equipotential rings around a point charge. Adjust charge and see how potential falls off as 1/r.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Electric potentialoutput Potential at distance r from the point charge | V | M·L²·T⁻³·I⁻¹ | -1000000 – 1000000 | |
| Source charge Point charge creating the potential | C | I·T | -0.00001 – 0.00001 | |
| Distance from charge Radial distance from the source charge | m | L | 0.01 – 2 |
Deep dive
Derivation
Integrate the electric field from infinity to r: V(r) = −∫_∞^r E·dr = −∫_∞^r (kq/r²)dr = kq/r. The negative sign ensures V decreases in the direction of E for positive charges.
Experimental verification
Measured using electrometers and potentiometers since the 19th century. Modern atomic-scale verification via scanning tunneling microscopy, which directly maps the potential landscape on surfaces.
Common misconceptions
- Potential is a scalar, not a vector — it has no direction
- Zero potential does not mean zero field; the field is the gradient of V
- Potential at a point depends on all charges in the universe, not just nearby ones
Real-world applications
- Voltage measurements in every electrical circuit
- Cardiac defibrillator energy calculations
- Electron volt as an energy unit in particle physics
- Electrostatic potential mapping in semiconductor design
Worked examples
Potential near a point charge
Given:
- q:
- 0.000008
- r:
- 0.5
Find: V
Solution
V = k_e × q / r = 8.9875×10⁹ × 8×10⁻⁶ / 0.5 = 1.438×10⁵ V = 143.8 kV
Work to bring a charge from infinity
Given:
- Q:
- 0.000005
- q:
- 0.000002
- r:
- 0.1
Find: Work W to bring q from infinity to r
Solution
W = qV = q × k_e × Q / r = 2×10⁻⁶ × 8.9875×10⁹ × 5×10⁻⁶ / 0.1 = 0.899 J
Scenarios
What if…
- scenario:
- What if the charge is negative?
- answer:
- Potential is negative at all points. V = −143.8 kV for −8 μC at 0.5 m. A positive test charge would gain kinetic energy falling toward it.
- scenario:
- What if you halve the distance?
- answer:
- Potential doubles: V = 287.6 kV. Unlike the field (1/r²), potential only falls as 1/r — slower decay.
- scenario:
- What if two equal charges create the potential?
- answer:
- Potentials are scalars and simply add: V_total = V₁ + V₂. No vector addition needed — this is the power of working with potentials.
Limiting cases
- condition:
- r → ∞
- result:
- V → 0
- explanation:
- Potential is defined to be zero at infinity.
- condition:
- r → 0
- result:
- V → ±∞
- explanation:
- Potential diverges at the location of a point charge.
- condition:
- q → 0
- result:
- V → 0
- explanation:
- No source charge means no potential.
Context
George Green · 1828
Green introduced the potential function in his self-published essay, unifying the mathematics of gravitational and electrostatic theory. His work was rediscovered by Lord Kelvin decades later.
Hook
What determines the 'voltage' at a point in space near a charge?
Calculate the electric potential at 0.5 m from a +8 μC point charge.
Dimensions: [V] = [k_e]·[q]·[r]⁻¹ → (N·m²·C⁻²)(C)(m⁻¹) = N·m/C = J/C = V ✓
Validity: Valid for point charges in vacuum with potential referenced to infinity. For continuous distributions, integrate contributions. In conductors, V is constant throughout the bulk.