Find the equal charge magnitude on two identical balls hanging symmetrically from a common point. The calculator also gives their center separation, horizontal electric force and thread tension.
Equilibrium and charge formula
For thread length L and angle θ from vertical, separation r = 2L sin(θ). Vertical balance gives T cos(θ) = mg; horizontal balance gives Fₑ = mg tan(θ).
Coulomb’s model is Fₑ = kₑq²/(εᵣr²), so |q| = r√(Fₑεᵣ/kₑ). Mass m is in kg, length r in m, force in N and charge in C. The calculator uses kₑ = 8.9875517862 × 10⁹ N·m²/C². Blank constant fields use g = 9.80665 m/s² and εᵣ = 1.
Example
Each ball has mass 10 g; each thread is 0.5 m long and makes 30° with vertical. With standard gravity and εᵣ = 1, r = 0.5 m and Fₑ ≈ 0.056619 N. The charge magnitude is approximately 1.254958 µC per ball.
Frequently asked questions
Is the angle between the two threads?
No. Enter each thread’s angle from vertical. In a symmetric arrangement, the total angle between threads is twice this value.
Does the result determine positive or negative charge?
No. Repulsion means the equal charges have like signs, but either both-positive or both-negative charges can fit the magnitude calculation.
Why are 0° and 90° excluded?
At 0°, the assumed separation collapses; at 90°, finite vertical equilibrium cannot be maintained. The model also assumes small balls relative to their separation, massless threads and negligible charge redistribution.
References
