Question 517: Electrostats & Capacitance

Question

Two electrons each are fixed at a distance '2d'. A third charge proton placed at the midpoint is displaced slightly by a distance x (x << d) perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency : (m = mass of charged particle)

Accepted Answer

The arrangement of charges is shown below
As we know that,
Coulomb's force between two charges. i.e., q
1
and q
2
,
$$F = {1 \over {4\pi {\varepsilon _0}}}{{{q_1}{q_2}} \over {{r^2}}} = {1 \over {4\pi {\varepsilon _0}}}{{{q_1}{q_2}} \over {({d^2} + {x^2})}}$$ ..... (i)
Here, $${q_1} = {q_2} = q$$
Force in SHM, $$F = m{\omega ^2}x$$ ...... (ii)
Since, in order to have SHM +q should move downwards and force responsible for this will be only
$$F' = F\sin 	heta  + F\sin 	heta  = 2F\sin 	heta $$ ..... (iii)
Using Eqs. (ii) and (iii), we get
$$2F\sin 	heta  = m{\omega ^2}x$$
$$ \Rightarrow {2 \over {4\pi {\varepsilon _0}}}{{{q^2}} \over {({d^2} + {x^2})}}\sin 	heta  = m{\omega ^2}x$$
$$ \Rightarrow {2 \over {4\pi {\varepsilon _0}}}{{{q^2}} \over {({d^2} + {x^2})}}.{x \over {{{({d^2} + {x^2})}^{1/2}}}} = m{\omega ^2}x$$
$$ \Rightarrow \omega  = {\left( {{1 \over {2\pi {\varepsilon _0}}}{{{q^2}} \over {{{({d^2} + {x^2})}^{3/2}}m}}} ight)^{1/2}}$$
As, $$x <  < d$$
$$	herefore$$ $$\omega  = {\left( {{1 \over {2\pi {\varepsilon _0}}}{{{q^2}} \over {m{d^3}}}} ight)^{1/2}}$$

Answer

$${\left( {{{2{q^2}} \over {\pi {\varepsilon _0}m{d^3}}}} \right)^{{1 \over 2}}}$$

Answer

$${\left( {{{{q^2}} \over {2\pi {\varepsilon _0}m{d^3}}}} \right)^{{1 \over 2}}}$$

Answer

$${\left( {{{2\pi {\varepsilon _0}m{d^3}} \over {{q^2}}}} \right)^{{1 \over 2}}}$$

Answer

$${\left( {{{\pi {\varepsilon _0}m{d^3}} \over {2{q^2}}}} \right)^{{1 \over 2}}}$$

Electrostats & Capacitance

Two electrons each are fixed at a distance '2d'. A third charge proton placed at the midpoint is displaced slightly by a distance x (x << d) perpendicular to the line joining the two fixed charges. Proton will execute simple harmonic motion having angular frequency : (m = mass of charged particle)

Choose the correct answer: