Capacitors in Series & Parallel

Capacitors are always connected in series and parallel or both.

Parallel Connection

This is when capacitors are arranged side by side with the positive terminal joined to a common junction.

The total charge stored in a capacitor, 

Q ₌ Q_{1 +} Q₂ ; Q_{1 =}C₁v; Q_{2 =} C₂v

Therefore, 

CV =C₁v + C₂v

Since V = V₁ =V₂, then

CV = C₁V+C₂V

or

C = C₁ + C₂    For parallel arrangement 

When capacitors are in parallel, 

1. The same voltage is applied across each capacitor
2. The charge Q is stored due to the current flowing and capacitance
3. The total charge stored by the capacitors equals sum of charge stored by each capacitor 

Series Connection

Capacitors are in series when they are joined end to end to form a line or loop.

1. The same current flows through each capacitor do the same change
2. Voltage through each capacitor depends on the capacitance of each capacitor
3. Total voltage is the sum of the voltage across the capacitor
4. Effective capacitance of the capacitor is given as

\( \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} \)

Example

Three capacitors of capacitance 2µF, 3µF and1µF are connected in a series in an electrical circuit. If a 500v dc source is connected across the capacitor. What is the total energy stored in the circuit.

Solution  

\( \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} \)

= \( \frac{1}{2} + \frac{1}{3} + \frac{1}{1} \)

= \( \frac{11}{6}\)

Therefore,

C = 0.55µF = 0.55 x 10^-6

Energy stored, E =  \( \frac{1}{2} \scriptsize cv^2 \)

=  \( \frac{1}{2} \scriptsize \; \times \; 0.55 \times 10^{-6} \times 500 \times 500 \)

=0.0625J

=6.25 10^-2 J