What do you understand by capacitors in series ? Derive a relation for equivalent capacitance of n capacitors connected in series.
Two or more capacitors are said to be connected in series if the right plate of one capacitor is connected to the left plate of other capacitor such that the magnitude of the charge (q) on each capacitor is same, when connected to a cell or battery.
Expression for Equivalent Capacitance:
Two capacitors of capacitance $C_1$ and $C_2$ connected in series. Let $V$ be the applied potential difference (through battery) across the two terminals A and B of the series combination of capacitors.
Total potential difference $V$ across the arrangement will be divided as $V_1$ and $V_2$ across each capacitor i.e. $C_1$ and $C_2$ respectively.
When capacitors are connected in series, potential difference across the combination is the sum of the potential differences across each capacitor.
i.e., $V = V_1 + V_2 \quad \dots \text{(i)}$
But
$V_1 = \frac{q}{C_1} \quad \text{and} \quad V_2 = \frac{q}{C_2}$
$V = \frac{q}{C_1} + \frac{q}{C_2} = q \left[ \frac{1}{C_1} + \frac{1}{C_2} \right] \quad \dots \text{(ii)}$
Let $C$ be the effective or equivalent capacitance of the series combination of the capacitors, then
$V = \frac{q}{C}$
Hence, eqn. (i) becomes
$\frac{q}{C} = q \left( \frac{1}{C_1} + \frac{1}{C_2} \right)$
$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} \quad \text{or} \quad C = \frac{C_1 C_2}{C_1 + C_2} \quad \dots \text{(iii)}$
where $C$ is capacitance of the series combination of the capacitors called equivalent capacitance or effective capacitance or net capacitance or total capacitance of the series combination of the capacitors.
or
If $n$ capacitors are connected in series, then
$\frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots + \frac{1}{C_n} \quad \dots \text{(iv)}$
$\frac{1}{C} = \sum_{i=1}^{n} \frac{1}{C_i} \quad \dots \text{(v)}$