Coulomb's Law — Question 6

Question 6

Suspended charged spheres couple electrical force to mechanical geometry

Use $k=8.99\times10^9\,\mathrm{N\,m^2/C^2}$; treat the specified charges as points in vacuum. Two tiny identical spheres of mass .0100 kg hang from the same point on insulating strings of length .500 m. Equal charges make both strings settle at $10.0^\circ$ from vertical. Use $g=9.80$ m/s$^2$ and neglect sphere size.

1. Derive the electrostatic force and charge magnitude from force balance and the actual sphere separation.

2. Derive the small-angle scaling of charge with angle in radians, and explain why using string length as the charge separation is incorrect.

Question 6 - Solution

Suspended charged spheres couple electrical force to mechanical geometry

(a) Method and calculation

$$T\cos\theta=mg,\quad T\sin\theta=F_e\Rightarrow F_e=mg\tan\theta.$$$$r=2L\sin\theta\approx.173648\,\mathrm m,\quad |q|=r\sqrt{\frac{mg\tan\theta}{k}}\approx2.40748\times10^{-7}\,\mathrm C.$$

(b) Interpretation and further calculation

$$\theta\ll1:\quad\sin\theta\simeq\theta,\quad\tan\theta\simeq\theta.$$$$|q|\simeq2L\sqrt{mg/k}\,\theta^{3/2}.$$The electrical force acts across the horizontal distance between spheres. That separation depends on the angle and tends to zero with it; the string length does not.

Physical check

The observation determines equal charge magnitudes and like signs, but cannot distinguish two positive charges from two negative charges.