Assuming mass m of the charged particle, find time period of oscillation

Question

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Answer 1

$E_{0} = 2 E_{1} c o s θ = \frac{q}{4 π ε _{0} ( x ^{2} + a ^{2} )} \frac{x}{x ^{2} + a ^{2}}$
$- q_{1} E = F = m f = \frac{- q _{1} q x}{4 π ε _{0} ( x ^{2} + a ^{2} ) ^{3 / 2}}$ as x << a, neglecting
x $^{2}$ as compared to acceleration, $f = \frac{- q _{1} q _{2} x}{4 π ε _{0} a ^{3} m}$
Comparing it with $f = ω^{2} x$
$ω \frac{q _{1} q}{4 π ε _{0} a ^{3} m}$ or $T = 2 π \frac{4 π ε _{0} a ^{3} m}{q _{1} q}$

Answer 2

E_{0} = 2 E_{1} c o s θ = \frac{q}{4 π ε _{0} ( x ^{2} + a ^{2} )} \frac{x}{x ^{2} + a ^{2}}

- q_{1} E = F = m f = \frac{- q _{1} q x}{4 π ε _{0} ( x ^{2} + a ^{2} ) ^{3 / 2}}

as x << a, neglecting
x

^{2}

as compared to acceleration,

f = \frac{- q _{1} q _{2} x}{4 π ε _{0} a ^{3} m}

Comparing it with

f = ω^{2} x

ω \frac{q _{1} q}{4 π ε _{0} a ^{3} m}

or

T = 2 π \frac{4 π ε _{0} a ^{3} m}{q _{1} q}