Question

Three identical metal plates with large surface areas are kept parallel to each other as shown in the following figure. The leftmost plate is given a charge Q, the rightmost a charge −2Q and the middle one is kept neutral. Find the charge appearing on the outer surface of the rightmost plate.

Answer 1

Consider the Gaussian surface as shown in the figure.

Let the charge on the outer surface of the left-most plate be q. Thus, the charges on the plates are distributed as shown in the following diagram.

The net field at point P due to all the induced charges must be zero, as it is lying inside the metal surface.

Let the surface area of the plates be A.

Electric field at point P due to the charges on plate X:-

Due to charge (+ Q -q) is `("Q" -"q")/(2"A"∈_0)`  in the right direction

Due to charge (+q) is `"q"/(2"A"∈_0)` in the right direction

Electric field at point P due to charges on plate Y:-

Due to charge (-q) is `"q"/(2"A"∈_0)` in the right direction

Due to charge (+q) is `"q"/(2"A"∈_0)` in the left direction.

Electric field at point P due to charges on plate Z:-

Due to charge (-q) is `"q"/(2"A"∈_0)` in the right direction.

Due to charge (-2Q+q) is `(2"Q"-"q")/(2"A"∈_0)` in the right direction

The net electric field at point P:-

`("Q"-"q")/(2"A"∈_0)+"q"/(2"A"∈_0) -"q"/(2"A"∈_0) - "q"/(2"A"∈_0) + "q"/(2"A"∈_0) + (2"Q"-"q")/(2"A"∈_0) = 0`

`("Q"-"q")/(2"A"∈_0) + (2"Q" -"q")/(2"A"∈_0) = 0`

`Q - q + 2Q - q = 0`

`3Q - q = 0`

`"q" =(3"Q")/2`

Thus, the charge on the outer plate of the right-most plate = `-2"Q" +"q" = -2"Q"+ (3"Q")/2 = -"Q"/2`