A force of 10N acts on a charged particle placed between two plates...

Electric field due to each plate, ${E}_{1} = {E}_{2} = \dfrac{\sigma}{2{ε}_{0}} = \dfrac{Q}{2A{ε}_{0}}$

Net electric field between the plates, ${E}_{net} = {E}_{1} + {E}_{2} = \dfrac{Q}{A{ε}_{0}}$

Force on charged particle between the plates, ${F}_{1} = q{E}_{net} = \dfrac{qQ}{A{ε}_{0}} = 10N$

Now, in second case, the net electric field, $E = \dfrac{\sigma}{2{ε}_{0}} = \dfrac{Q}{2A{ε}_{0}} = 5N$

Force on charged particle, ${F}_{2} = qE = \dfrac{qQ}{2A{ε}_{0}} = 5N$