Two identical positive charge Q each are fixed at a distance of '2a' apart from each other. Another point charge q₀ with mass 'm' is placed at midpoint between two fixed charges. For a small displacement along the line joining the fixed charges, the charge q₀ executes SHM. The time period of oscillation of charge q₀ will be:

$F_R=\left\{\frac{kQ{q}_0}{{\left(a-x\right)}^2}-\frac{kQ{q}_0}{{\left(a+x\right)}^2}\right\}$

$=-kQ{q}_0\left\{\frac{a^2+x^2+2ax-a^2-x^2+2ax}{\left(a^2-x^2\right)}\right\}$

$F_R=-\frac{kQ{q}_0\left(4ax\right)}{{\left(a^2-x^2\right)}^2}$

$T=2\pi \sqrt[]{\frac{a^{3}m}{4kQ{q}_0}=\sqrt[]{\frac{4{\pi }^2{a}^{3}m*4\pi {\epsilon }_0}{4Q{q}_0}}}=\sqrt[]{\frac{4{\pi }^3{\epsilon }_{0}m{a}^3}{Q{q}_0}}$

$F_R=\left\{\frac{kQ{q}_0}{{\left(a?x\right)}^2}?\frac{kQ{q}_0}{{\left(a+x\right)}^2}\right\}$

$=?kQ{q}_0\left\{\frac{a^2+x^2+2ax?a^2?x^2+2ax}{\left(a^2?x^2\right)}\right\}$

$F_R=?\frac{kQ{q}_0\left(4ax\right)}{{\left(a^2?x^2\right)}^2}$

$T=2?\sqrt[]{\frac{a^{3}m}{4kQ{q}_0}=\sqrt[]{\frac{4{?}^2{a}^{3}m*4?{?}_0}{4Q{q}_0}}}=\sqrt[]{\frac{4{?}^3{?}_{0}m{a}^3}{Q{q}_0}}$

$F_R=\left\{\frac{kQ{q}_0}{{\left(a?x\right)}^2}?\frac{kQ{q}_0}{{\left(a+x\right)}^2}\right\}$

$=?kQ{q}_0\left\{\frac{a^2+x^2+2ax?a^2?x^2+2ax}{\left(a^2?x^2\right)}\right\}$

$F_R=?\frac{kQ{q}_0\left(4ax\right)}{{\left(a^2?x^2\right)}^2}$

$T=2?\sqrt[]{\frac{a^{3}m}{4kQ{q}_0}=\sqrt[]{\frac{4{?}^2{a}^{3}m*4?{?}_0}{4Q{q}_0}}}=\sqrt[]{\frac{4{?}^3{?}_{0}m{a}^3}{Q{q}_0}}$

$F_R=\left\{\frac{kQ{q}_0}{{\left(a?x\right)}^2}?\frac{kQ{q}_0}{{\left(a+x\right)}^2}\right\}$

$=?kQ{q}_0\left\{\frac{a^2+x^2+2ax?a^2?x^2+2ax}{\left(a^2?x^2\right)}\right\}$

$F_R=?\frac{kQ{q}_0\left(4ax\right)}{{\left(a^2?x^2\right)}^2}$

$T=2?\sqrt[]{\frac{a^{3}m}{4kQ{q}_0}=\sqrt[]{\frac{4{?}^2{a}^{3}m*4?{?}_0}{4Q{q}_0}}}=\sqrt[]{\frac{4{?}^3{?}_{0}m{a}^3}{Q{q}_0}}$