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

(C) $\left(\frac{q^{2}}{2 \pi \varepsilon_{0} m d^{3}}\right)^{\frac{1}{2}}$

(A) $\left(\frac{2 q^{2}}{\pi \varepsilon_{0} m d^{3}}\right)^{\frac{1}{2}}$

(B) $\left(\frac{2 \pi \varepsilon_{0} m d^{3}}{q^{2}}\right)^{\frac{1}{2}}$

(D) $\left(\frac{2 \pi \varepsilon_{0} m d^{3}}{2 q^{2}}\right)^{\frac{1}{2}}$

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Elasticity & SHM

Two electrons each are fixed at a distance '

2 d

'. 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)

(A) $(\frac{2 q ^{2}}{π ε _{0} m d ^{3}})^{\frac{1}{2}}$
(B) $(\frac{2 π ε _{0} m d ^{3}}{q ^{2}})^{\frac{1}{2}}$
(C) $(\frac{q ^{2}}{2 π ε _{0} m d ^{3}})^{\frac{1}{2}}$ ✓ Correct
(D) $(\frac{2 π ε _{0} m d ^{3}}{2 q ^{2}})^{\frac{1}{2}}$

Answer: (C)

Solution

Each electron sits a distance

d

from the midpoint. When the proton is displaced by

x

perpendicular to the line joining them, each electron attracts it with a force of magnitude

\frac{k q ^{2}}{d ^{2} + x ^{2}}

, where

k = \frac{1}{4 π ε _{0}}

. The components along the line of the electrons cancel by symmetry, while the components along the displacement add and point back towards the midpoint:

F = - 2 \frac{k q ^{2}}{d ^{2} + x ^{2}} \cdot \frac{x}{d ^{2} + x ^{2}}

For

x ≪ d

the denominator becomes

d^{3}

, leaving

F = - \frac{2 k q ^{2}}{d ^{3}} x

, a restoring force proportional to the displacement. Comparing with

F = - m ω^{2} x

ω^{2} = \frac{2 q ^{2}}{4 π ε _{0} m d ^{3}} = \frac{q ^{2}}{2 π ε _{0} m d ^{3}}

so the angular frequency is

(\frac{q ^{2}}{2 π ε _{0} m d ^{3}})^{\frac{1}{2}}

JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Single correct option

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JEE Main · 2021 · 24 Feb 2021 · Shift 2 · Elasticity & SHM

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