There are n large parallel plate conductors carrying charge `Q_1,Q_2,………..Q_n` respectively.

i. Find the charge induced at surface A.
ii. Find the charge induced at surface B.
iii. If the left conductor is earthed, find the magnitude of charge flowing from plate to earth.
iv. If any conductor is earthed, find the magnitude of charge flowing from plate to earth.

Explanation

While solving the problems of multiple plates
placed parallel to each
other, the following points
should be keep in mind.
`*` Charges appearing
on the outer surfaces should be equal in
magnitudes as well as
sign.
`*` Charges appearing on the facing
surfaces should be
equal in magnitude but opposite in sign.
If we connect any plate (innermost or outermost) with
earth, the charge appearing on the outer surfaces will be
zero.
`*` The charges of the plates other than those connected with
earth remain conserved.
`*` The charges of the plates other than those connected with
earth remain conserved.
i. Charge on outermost surface A is half the summation of charges on both systems. So

`Q_(A) = (Q_1+sum_(i=2)^(i=n) Q_i)/2 = 1/2 sum_(i=1)^(i=n)Q_i`
ii. Charge on surface B is half the differenec of charge on both systems. So,
`Q_B = -(Q_1-sum_(i=2)^(i=n)Q_i)/2 = -1/2 [Q_1 - sum_(i-2)^(i=n)Q_i]`
iii. If the outer plate is connected with earth, the charge on the outermost surfaces will be zero.
Initial charge in plate 1 `=Q_1`
Final charge in plate 1 = `-sum_(i=2)^(i=n)Q_i`
Charge x flowing from plate 1 to earth is
`-sum_(i=2)^(i=n)Q_i+x = Q_1`
or `x = Q_1 + sum_(i =2)^(i=n)Q_i = sum_(i=1)^(i=n)Q_i`

Let the second plate from the left be earthed. The
charge on the outermost surfaces of the plate should be zero.
Initial charge in plate `2 = Q_2`
Final charge in plate `2 = -Q_1-sum_(i=3)^(i=n)Q_i`
Let the charge flowing from plate to earth be y.

`:. -Q_1 - sum_(i=3)^(i=n)Q_i+y= Q_2`
or `y = Q_1+Q_2 + sum_(i=3)^(i=n)Q_i = sum_(i=1)^(i=n)Q_i`
Hence, if we connect any plate with earth, the charge
flowing from the plate to earth will be ` sum_(i=1)^(i=n)Q_i`.