Question

Two particles A and B, each arrying a charge Q, are held fixed with a separation d between them. A particle C having mass m and charge q is kept at the middle point of the line AB. (a) if it is displaced through a distane x perpendicular to AB, what would be the electric forc experienced by it (b) Assuming x<<d, show that this force is proportional to x. (c) Under what conditions will the particle C execute simple harmonic motion if it is released after such a small displacement? Find the time perio of the oscillations if these conditions are satisfied.

Answer 1

(a) From previous solution.
Net force acting
$=\frac{1}{2\pi {ϵ}_0}\frac{q\theta x}{{[{\left(\frac{d}{2}\right)}^2+x^2]}^{\frac{8}{2}}}\left(b\right)x<<d$
The force $=\frac{1}{2\pi {ϵ}_0}\frac{q\theta x}{{\left(\frac{d}{2}\right)}^3}=\frac{4}{\pi {ϵ}_0}\frac{q\theta x}{d^3}$
(c) Force$=m{w}^{2}x$
$\frac{4}{\pi {ϵ}_0}\frac{q\theta x}{d^2}=m{\left(\frac{2\pi }{t}\right)}^2.xor{T}^2=\frac{m{\pi }^3{ϵ}_0{d}^3}{\theta q}orT={\left[\frac{m{\pi }^3{ϵ}_0{d}^3}{\theta q}\right]}^{\frac{1}{2}}$