Problem — Electric pendulum

Problem

Electric pendulum. A ball of mass $5\ \text{g}$ with charge $+1\ \mu\text{C}$ hangs from a thread in a uniform horizontal electric field $E = 5\cdot 10^3\ \text{V/m}$. At what angle does the thread tilt from the vertical?

Solution

Equilibrium. The ball is acted on by its weight $mg$ (vertical), the horizontal electric force $F_{\text{el}} = qE$ and the tension in the thread. At equilibrium the tension balances the resultant of weight and electric force, so the angle $\theta$ from the vertical satisfies:

$\tan\theta = \frac{F_{\text{el}}}{mg} = \frac{qE}{mg}$

Numerical substitution. With $q = 1\cdot 10^{-6}\ \text{C}$, $E = 5\cdot 10^3\ \text{V/m}$, $m = 5\cdot 10^{-3}\ \text{kg}$, $g = 9{,}8\ \text{m/s}^2$:

$\tan\theta = \frac{1\cdot 10^{-6} \cdot 5\cdot 10^3}{5\cdot 10^{-3} \cdot 9{,}8} = \frac{5\cdot 10^{-3}}{0{,}049} \approx 0{,}1020$

Angle.

$\theta = \arctan(0{,}1020) \approx 5{,}82^\circ$

$\ev{\theta \approx 5{,}82^\circ}$

Linked atoms

Topics: Campo elettrico e potenziale Concepts: Campo elettrico Objects: Pendolo semplice