The thickness of a flat sheet of metal foil is $$d,$$ and its area is $$S.$$ A charge $$q$$ is located at a distance $$\ell $$ from the centre of the sheet such that $$d < < \sqrt S < < \ell .$$    Determine the force $$F$$ with which the sheet is attracted to the charge $$q,$$ assuming that the straight line connecting the charge to the centre of the sheet is perpendicular to the surface of the sheet.
(approximately)

Solution :
Since the sheet is metallic, the charges must be redistributed over its surface so that the field in the bulk of the sheet is zero. In the first approximation, we can assume that this distribution is uniform and has density $$ - \sigma $$ and $$\sigma $$ over the upper and the lower surface respectively of the sheet. According to the superposition principle, we obtain the condition for the absence of the field in the bulk of the sheet :
$$\frac{q}{{4\pi {\varepsilon _0}{\ell ^2}}} - \frac{\sigma }{{{\varepsilon _0}}} = 0$$
Let us now take into consideration the non uniformity of the field produced by the point charge since it is the single cause of the force $$F$$ of interaction. The upper surface of the sheet must be attracted with a force $$\frac{{\sigma Sq}}{{4\pi {\varepsilon _0}{\ell ^2}}},$$  while the lower surface must be repelled with a force $$\frac{{\sigma Sq}}{{4\pi {\varepsilon _0}{{\left( {\ell + d} \right)}^2}}}.$$
Consequently, the force of attraction of the sheet to the charge is
$$F = \frac{{\sigma Sq}}{{4\pi {\varepsilon _0}{\ell ^2}}}\left[ {1 - \frac{1}{{{{\left( {1 + \frac{d}{\ell }} \right)}^2}}}} \right] \approx \frac{{{q^2}Sd}}{{8{\pi ^2}{\varepsilon _0}{\ell ^5}}}$$