Three parallel metalic plates each of area `A` are kept as shown in figure and charges `q_1,q_2` and `q_3` are given to them. Find the resulting charge distribution on the six surfaces, neglecting edge effects as usual,

Explanation

The plate separations do not affect the distribution of charge in this problem.
In the figure
`q_0=q_1-q_a,q_d=q_2-q_c`
`and q_f=q_3-q_e`
electric field at point P is zero because this point is lying inside a conductor.
`E=0`

At `P` charge `q_a` will give an electric field towards right. All other charges `q_0, q_c.`.........etc. will give electric field towars left. So,
`1/(2Aepsilon_0)[q_a-(q_1-q_1)-q_c-(q_2-q_c)-q_e-(q_3-q_e)]=0`
or `2q_a-q_1-q_2-q_3=0`
or `q_a=(q_1+q_2+q_3)/2`
similarly the conditionn
`E_R=0`
will give the result
`q_f=(q_1+q_2+q_3)/2`
from here we may conclude that half of the sum of all charges appears on each of the two outermost surfaces of the system of plates.
Further we have a condition.
`E_Q=0`
`1/(2Aepsilon_0)[q_q+(q_1-q_a)+q_c-(q_2-q_c)-q_e-(q_3-q_e)]=0`
or `q_1+2q_c-q_2-q_3=0`
`:. q_c=(q_2+q_3-q_1)/2`
`q_b=q_1-q_a=(q_1-q_2-q_3)/2=-q_c`
similarly, we can show that
`q_d=-q_e`
from here we can find another important resultant that the pairs of opposite surfaces like `b,c`, and `d,e` carry equal and opposite charges.