Answer to Question #310915 in Electric Circuits for Nataša

Question

On threads of equal lengths of 50 cm and

negligible masses, about the same point

two balls of equal mass were hung

0.3 g each. When the balls are charged

equal amounts of the same

charges are reflected five times

less distance than length

the end. Calculate the charge of one

balls.

Answer 1

Explanations & Calculations

• The electrostatic force between the charges according to Coulomb's law when there is a $\small d$ separation between them is

$\qquad\qquad \begin{aligned} \small F&=\small k\frac{Q\times Q}{d^2}\\ \small F&=\small k.\frac{Q^2}{d^2}\cdots\cdots(1) \end{aligned}$

• Now, from the equilibrium of a single ball-thread system if the angular displacement measured from the vertical is $\small \theta$ is

$\qquad\qquad \begin{aligned} \small \uparrow\\ \small T\cos\theta&=\small mg\\ \to\\ \small T\sin\theta&=\small F\\ \\ \small F&=\small mg.\tan\theta \end{aligned}$

• This into (1) yeilds

$\qquad\qquad \begin{aligned} \small k.\frac{Q^2}{d^2}&=\small mg.\tan\theta\\ \small Q&=\small \pm \,d\sqrt{\frac{mg\tan\theta}{k}}\cdots(Answer)\\ \\ \small \sin\theta&=\small \frac{d/2}{l}\\ \small \theta&=\small \sin^{-1}(d/2l) \end{aligned}$

• Once you know how much the balls are separated you can calculate the angle and then the tangent of it and finally the charge of a ball.