Three capacitors `C_1, C_2, C_3` are Connected in series. Derive an expression for the equivalent capacitance.

Explanation

Capacitors are said to be.connected in series when these are joined one after the other in such a way that charge across each capacitor remains same.
Equivalent Capacitance of Capacitors in Series Combination.
Let we have three capacitors `C_(1), C_(2)` and `C_3`. In series combination charge on the plates of each capacitor remains i.e., q (say).
Let `V_(1), V_(2), V_(3)`= Potential differences across the plates of capacitors `C_(1), C_(2)` and `C_3` respectively.
V= Total potential difference across the combination
When capacitors are connected in series, potential difference across the combination is the sum of the potential differences across each .capacitor.
i.e., `V =V_(1) + V_(2) + V_(3)`.......(i)
But, `V_(1) =q/C_(1), V_(2) =q/C_(2), V_(3) = q/C_(3)` and `V = q/C`
where C - Equivalent capacitance of capacitors in series combination.
Put the values of V, `V_(1), V_(2)` and `V_(3)` in eqn. (i)

`q/C=q/C_(1) +q/C_(2) + q/C_(3)`
`q/C =q(1/C_(1) + 1/C_(2) + 1/C_(3))`
`rArr 1/C = 1/C_(1) + 1/C_(2) + 1/C_(3)`
So in series combination, reciprocal of equivalent capacitance of capacitors in series combination is equal to the sum of reciprocals of individual capacitances of the capacitors present in the combination. Note : Series combination of capacitors is preferred when we need smaller capacitance.