Question

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5. A point charge q is placed at a distance a from an infinite conducting plate which has no net charge. Calculate the surface charge density of the boundary of the conducting plate facing this point charge. The charge density is not a constant. You can choose a convenient coordinate system on the conducting surface such that the point closest to the point charge is the origin. Then integrating over the whole surface of the conducting plate facing the point charge and verify that the total charge is -q.

Answer 1

"Given that charge = q distance = a Suppose the grounded condutind plate is in XY plane and a point charge ""q"" is along the z-axis at a distance of 'a' Let P(x, y) be any point of position vector the Potential at point P(x, y, z) due to charge +q V_+ q = 1 q/4 pi E_0 squareroot x^2 + y^2 + (z-a)^2 Potential at point P(x, y, z) due to image charge = -q V_-q = -q/4 pi E_0 (squareroot x^2 + y^2 + (z + a))^2 Total potential V = V_pq + V_-q V+ q/4 pi E_0 {1/ squareroot x^2 + y^2 + (z-a)^2-1/squareroot x^2 + y^2 + (z + a)^2} Now Electric field E = -v V x^2 + y^2 = p^2 E = (-p partial differential/ partial differential p-k partial differential/ partial differential 2)v implies E = {-p partial differential/partial differential p (1/ squareroot p^2 + (2-a)^2) + P partial differential/partial differential p(1/ squareroot p^2 + (z + a)^2)}"