We know, electric field due to uniformly charged infinite plane sheet,

$$E = {{{\sigma _0}} \over {2{\varepsilon _0}}}$$

perpendicularly directed away from the sheet for positive charge

So, $$\overrightarrow E = {{{\sigma _0}} \over {2{\varepsilon _0}}} \uparrow \theta $$

note : E is constant

Let length of dipole = d

so dipole moment, $$\overrightarrow p = qd \uparrow $$

We know,

Torque, $$\overrightarrow \tau = \overrightarrow p \times \overrightarrow E $$

$$\overrightarrow \tau = qd\widehat i \times {{{\sigma _0}} \over {2{\varepsilon _0}}}\widehat i$$

$$\widehat \tau = O$$ (as $$\widehat i \times \widehat i = 0$$)

force, on charge $$ - q,\overrightarrow {{F_1}} = - qE\widehat i$$

on charge $$ + q,\overrightarrow {{F_2}} = qE\widehat i$$

So net force on dipole, $$\overrightarrow F = \overrightarrow {{F_1}} + \overrightarrow {{F_2}} $$

$$ = - qE\widehat i + qE\widehat i$$

$$\overrightarrow F = O$$

We know, potential energy of dipole

$$u(O) = - pE\cos \theta $$

here, $$O = 0^\circ $$ (given)

So, $${u_{\min }} = - pE$$

Hence, option B is correct.