Series combination of condenser

“Series combination of condenser”:
 In this combination second plate of first condenser is joined with the first plate of second condenser and second plate of second condenser is joined with the first plate of third condenser and so on. The outer surface of second plate of last condenser is earthed and some amount of charge is given to the first plate of first condenser.

#Description:
In above figure shows the series combination of three condenser of capacitance C1, C2 and C3. In this combination there is only one part 2 flow of charge. Thus by the electrical induction equal amount of charge Q is produced on each plate of every condenser but the potential difference between their plates  are different i.e (V1 - V2), ( V2 - V3) and
(V3 - V4) .

We know that by the theory of the condenser:
C = Q/∆V
Hence, ∆V = Q/C

Potential difference between plates of first condenser:
V1 -V2  =  Q/C1

Potential difference between plates of second condenser:
V2- V3   =  Q/C2

Potential difference between plates of third condenser:
V3 - V4.  =  Q/C3

Adding all above eqn

(V1-V2)+(V2-V3)+(V3-V4) = Q/C1+Q/C2+Q/C3

V1 - V4 = Q( 1/C1 + 1/C2 + 1/C3 )------(1)

let the equivalent capacitance of all condenser are joined in series be C farad. Thus all these condenser are replaced by a single condenser of capacitance C . The charge on its plate be Q and potential difference between its plates be V1 - V4.

Thus by the theory of condenser
V1 - V4 = Q/C  --------(2)

Comparing eqn1 and eqn2

Q/C = Q(1/C1 + 1/C2 + 1/C3 )

[1/C =  1/C1 + 1/C2 + 1/C3 ]

This is the required expression for series combination of condenser.