= Solution
In electrostatic equilibrium, the <electric field> inside a <perfect conductor> is zero. Apply the <Faraday's law> $\oint\mathbf E\cdot d\mathbf l=0$ to a thin rectangular loop crossing the surface, with its long sides tangential and its short sides shrinking to zero. It yields continuity of the tangential <electric field>. Since its interior value is zero, \b[$E_x(0+)=E_y(0+)=0$]. Earthing fixes the <electric potential> at the plane to zero. For completeness, a thin <Gauss law> pillbox gives $E_z(0+)=\sigma/\varepsilon_0$; the normal component can be nonzero because of induced surface charge. These are the <electrostatic boundary conditions at a conductor>.
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