Library / Foundations: Physics / FPH-57
Foundations: Physics
Coulomb force between two point charges
Force between two small charged bodies, proportional to the product of the charges and falling with the square of their separation.
- Ref
- FPH-57
- Rev
- v1
- Field
- Foundations: Physics
- Result
- F [N]
q₁ and q₂ are the two charges in coulombs, signed: positive for positive charge, negative for negative. r is the centre-to-centre distance in metres. The constant 1/(4πε₀) is 8.988 × 10⁹ N·m²/C². A positive answer pushes the charges apart; a negative answer pulls them together.
Force between two small charged bodies, proportional to the product of the charges and falling with the square of their separation. A positive answer is repulsion (like charges); a negative answer is attraction (unlike charges).
Calculator
- Coulomb constant 1/(4πε₀)8.988×109 N·m²/C²
- Force if the distance were doubled0.3745 N
- Weight of the same force in grams152.7 g
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.
- The charges have the same sign, so they repel, with a force of 1.498 N. Microcoulomb charges a few tens of centimetres apart already give a noticeable force.
- Inverse-square law: doubling the distance cuts the force to 25 % of its value, and halving it multiplies the force by four.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (N) | Local step | Change in result |
|---|---|---|---|---|---|---|
| r Distance between the charges | 0.01 to 10 m | 0.001348 | 1,348 | 1,348 | +10% | −17.4 % |
| q2 Second charge (signed) | −0.001 to 0.001 C | −499.3 | 499.3 | 998.6 | +0.01 C | +333,000 % |
| q1 First charge (signed) | −0.001 to 0.001 C | −299.6 | 299.6 | 599.2 | +0.01 C | +200,000 % |
Sweep of Distance between the charges
| Distance between the charges (m) | Force (positive = repulsive) (N) |
|---|---|
| 0.01 | 1,348 |
| 0.02371 | 239.7 |
| 0.05623 | 42.63 |
| 0.1334 | 7.581 |
| 0.3162 | 1.348 |
| 0.7499 | 0.2397 |
| 1.778 | 0.04263 |
| 4.217 | 0.007581 |
| 10 | 0.001348 |
Sweep one variable
Variables and how each one contributes
Force (positive = repulsive)
What the equation gives you. Everything else feeds it.
First charge (signed)
The first charge in coulombs, with its sign. Raises the force in direct proportion; reversing its sign turns repulsion into attraction. Raising it by 10 % (to 5.5×10−6) moves the result by +10 %, from 1.498 to 1.648 N.
Second charge (signed)
The second charge in coulombs, with its sign. Raises the force in direct proportion; reversing its sign turns repulsion into attraction. Raising it by 10 % (to 3.3×10−6) moves the result by +10 %, from 1.498 to 1.648 N.
Distance between the charges
The distance between the charge centres, in m. Lowers the force with the square of the distance, so a small increase in r has twice the percentage effect. Raising it by 10 % (to 0.33) moves the result by −17.4 %, from 1.498 to 1.238 N.
Worked example with the values above
- Product of the charges: q₁q₂ = 5×10−6 × 3×10−6 = 1.5×10−11 C².
- Distance squared: r² = 0.3² = 0.09 m².
- F = 8.988 × 10⁹ × 1.5×10−11 / 0.09 = 1.49793 N.
Assumptions and limits
- Point charges, or spheres of uniform charge with r measured between their centres.
- Charges at rest in a vacuum or air; in another medium replace ε₀ by ε₀εr.
- No other charges nearby that would redistribute the charge.
Common mistakes
- Entering the distance in centimetres or the charge in microcoulombs without converting to metres and coulombs.
- Dropping the signs of the charges and losing the direction.
- Measuring the distance between surfaces instead of centres.
Statistics and uncertainty
Coulomb’s law is exact for point charges, so the uncertainty comes from the inputs. The force scales with 1/r², so a 1 % error in the distance gives a 2 % error in the force; measuring r between the charge centres is usually the biggest source of error for real bodies.
Where it is used
- Estimating electrostatic forces on dust, droplets and powder.
- Checking the scale of forces in capacitor plates and electrostatic actuators.
Maths you need
- Electric charge: the coulomb
- Inverse-square laws: force falls as 1/r²