An isolated conducting sheet of area A and carrying a charge Q is placed in a uniform electric field E, such that the electric field is perpendicular to the sheet and covers all the sheet. Find out the charges appearing on its two surfaces.

Explanation

Let there is x charge on left side of plate and Q-x charge on right side of plate
`:' E_(P)=0`
`:. x/(2Aepsi_(0))+E=(Q-x)/(2Aepsi_(0))` or `x/(A epsi_(0))=Q/(2Aepsi_(0))-E`
`:. x=Q/2-Eaepsi_(0)` and `Q-x=Q/2+Eaepsi_(0)`
So, charge on one side is `Q/2 -EAepsi_(0)` and other side `Q/2 + EAepsi_(0)`

Electric field due to conducting sheet at a point outside the sheet `=Q_("net")/(2Aepsi_(0))=Q/(2A epsi_(0))` and having direction away from plate `E_(1)` and `E_(2)` are electric field due to charges on left and right side of the plate respectively.
Now,
`E_("left")=E-E_(1)-E_(2)=E-((Q/2-EAepsi_(0))+(Q/2+EAepsi_(0)))/(2Aepsi_(0))=E-Q/(2Aepsi_(0)) ("towards right")`
`E_("right")=E+E_(1)+E_(2)=E+Q/(2Aepsi_(0)) ("towards right")`