Basic EM, Ez for you

First, we notice that the radius of the solenoid is smaller than its length and the distance between the coils are smaller than its radius. This means that we can think of the solenoid as the collection of many magnetic dipoles aligned in the vertical line. Because head and tail are canceling each other, the net effect is just two monopoles with a different sign but the same magnitude at each end of it.

The equivalent orientation of the solenoids implies that we can think of another system consisted of 4 monopoles (let's choose 2 + monopoles on above others on below) in each rectangular vertex with dimension $d \times L$ where $d$ is the separation distance. The magnetic field by a monopole can be written as

$\vec {B} = \frac {\mu_0 q_m}{4\pi r^3} \vec {r}$

where $q_m$ is the magnitude of the monopole that depends on the current and internal geometry of the solenoid.

The force exerted on a monopole is given as

$\vec {F} = q_m \vec {B}$

So, by vector analysis, one obtains that this type of force is REPULSION and the magnitude is

$F = \frac {\mu_0 q_m}{2\pi}[\frac {1}{d^2} - \frac{d}{(d^2+L^2)^{3/2}}]$

By substitution of two cases and take the ratio, we get

$x = \frac {1/4 - \frac {2}{5^{3/2}}}{1- \frac {1}{2^{3/2}}} \approx 0.110$