A point charge q is at distance a from an infinite grounded conducting plane. It is slowly moved normally away from the plane until its distance becomes 2a. The external work required is:
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A point charge q is at distance a from an infinite grounded conducting plane. It is slowly moved normally away from the plane until its distance becomes 2a. The external work required is:

G1 Physics · Accepted Answer

Using the image charge -q, the electrostatic energy of the real charge–plane system at distance x is U(x)=-q²/(16πε₀x). Therefore W e x t = U ( 2 a ) − U ( a ) = − q 2 / ( 32 π ε 0 a ) + q 2 / ( 16 π ε 0 a ) = q 2 / ( 32 π ε 0 a ) . W_ext=U(2a)-U(a)=-q²/(32πε₀a)+q²/(16πε₀a)=q²/(32πε₀a). W e ​ x t = U ( 2 a ) − U ( a ) = − q 2 / ( 32 π ε 0 ​ a ) + q 2 / ( 16 π ε 0 ​ a ) = q 2 / ( 32 π ε 0 ​ a ) .