A Problem From IE Irodov

The method of images gives us the effective charge arrangement shown above. The resulting electric force on the top left charge is:

$\large{\vec{F} = - \frac{q^2}{4 \pi \, \epsilon_0 \, \ell^2} \, \hat{\jmath} + \frac{q^2}{4 \pi \, \epsilon_0 \, \ell^2} \, \hat{\imath} + \frac{q^2}{4 \pi \, \epsilon_0 \, 2 \ell^2} \, \Big [ - \frac{1}{\sqrt{2}} \hat{\imath} + \frac{1}{\sqrt{2}} \hat{\jmath} \Big ] \\ = \frac{q^2}{4 \pi \, \epsilon_0 \, \ell^2} \, \Big [ \Big (1 - \frac{1}{2 \sqrt{2}} \Big ) \hat{\imath} + \Big (-1 + \frac{1}{2 \sqrt{2}} \Big ) \hat{\jmath} \Big ] }$

Magnitude of electric force:

$\large{|\vec{F}| = \frac{q^2}{4 \pi \, \epsilon_0 \, \ell^2} \, \sqrt{2} \, \Big(1 - \frac{1}{2 \sqrt{2}} \Big ) \\ = \frac{q^2}{4 \pi \, \epsilon_0 \, \ell^2} \, \Big(\sqrt{2} - \frac{1}{2 } \Big ) \\ = \frac{q^2}{8 \pi \, \epsilon_0 \, \ell^2} \, \Big(2 \sqrt{2} - 1 \Big )}$

Now for the electric field strength between the charges. The field is purely in the horizontal direction. The contribution from the two real charges is:

$2 \frac{q}{4 \pi \, \epsilon_0 \, (\frac{\ell}{2})^2} = \frac{2 q}{\pi \, \epsilon_0 \, \ell^2}$

The contribution from the two fictitious charges is:

$-2 \frac{q}{4 \pi \, \epsilon_0 \Big (\frac{\sqrt{5}}{2 } \ell \Big )^2} \frac{\ell/2}{\sqrt{5} \ell / 2} = -\frac{2 q}{5 \sqrt{5} \, \pi \epsilon_0 \, \ell^2}$

Summing them together results in:

$|\vec{E}| = \frac{2 q}{\pi \epsilon_0 \, \ell^2} \Big (1 - \frac{1}{5 \sqrt{5}} \Big )$