Capacitors in Series & Parallel

The combined capacitance of capacitors in a circuit depends on whether they are connected in series or parallel

Capacitors in Series

Consider two parallel plate capacitors C₁and C₂ connected in series, with a potential difference (p.d) V across them

Potential Difference of Capacitors Connected in Series

Capacitors in series, downloadable AS & A Level Physics revision notes

Capacitors connected in series have different p.d across them but have the same charge

In a series circuit, p.d. is shared between all the components in the circuit
- Therefore, if the capacitors store the same charge on their plates but have different p.ds, the p.d across C₁ is V₁ and across C₂ is V₂
The total potential difference V is the sum of V₁ and V₂

V = V₁ + V₂

Rearranging the capacitance equation for the p.d. V means V₁ and V₂ can be written as:

$V_{1} = \frac{Q}{C_{1}}$ and $V_{2} = \frac{Q}{C_{2}}$

Where the total p.d. V is defined by the total capacitance

$V = \frac{Q}{C_{t o t a l}}$

Substituting these into the equation V = V₁ + V₂ equals:

$\frac{Q}{C_{t o t a l}} = \frac{Q}{C_{1}} + \frac{Q}{C_{2}}$

Since the current is the same through all components in a series circuit, the charge Q is the same through each capacitor and cancels out
Therefore, the equation for the combined capacitance of capacitors in series is:

$\frac{1}{C_{t o t a l}} = \frac{1}{C_{1}} + \frac{1}{C_{2}} + \frac{1}{C_{3}} . . . . .$

Capacitors in Parallel

Consider two parallel plate capacitors C₁ and C₂ connected in parallel, each with p.d, V

Potential Difference of Capacitors Connected in Parallel

Capacitors in parallel, downloadable AS & A Level Physics revision notes

Capacitors connected in parallel have the same p.d across them, but different charge

Since the current is split across each junction in a parallel circuit, the charge stored on each capacitor is different
Therefore, the charge on capacitor C₁ is Q₁ and on C₂ is Q₂
The total charge Q is the sum of Q₁ and Q₂

Q = Q₁ + Q₂

Rearranging the capacitance equation for the charge Q means Q₁ and Q₂ can be written as:

Q₁ = C₁V and Q₂ = C₂V

Where the total charge Q is defined by the total capacitance:

Q = C_totalV

Substituting these into the Q = Q₁ + Q₂ equals:

C_totalV = C₁V + C₂V = (C₁+ C₂) V

Since the p.d is the same through all components in each branch of a parallel circuit, the p.d V cancels out
Therefore, the equation for the combined capacitance of capacitors in parallel is:

C_total = C₁+ C₂ + C₃ ...

Worked example

Three capacitors with capacitance of 23 μF, 35 μF and 40 μF are connected as shown below. WE Capacitors in Series & Parallel question image, downloadable AS & A Level Physics revision notes Calculate the total capacitance between points A and B.

Answer:

Step 1: Calculate the combined capacitance of the two capacitors in parallel

Capacitors in parallel: C_total = C₁+ C₂ + C₃ …

C_parallel= 23 + 35 = 58 μF

Step 2: Connect this combined capacitance with the final capacitor in series

Capacitors in Series: $\frac{1}{C_{total}} = \frac{1}{C_{1}} + \frac{1}{C_{2}} + \frac{1}{C_{3}} . . . . .$

$\frac{1}{C_{t o t a l}} = \frac{1}{58} + \frac{1}{40} = \frac{49}{1160}$

Step 3: Rearrange for the total capacitance

$C_{t o t a l} = \frac{1160}{49} = 23.673 . . . = 24 μ F (2 s . f .)$

Exam Tip

Make sure you learn the equation for the combined capacitance of capacitors in both series and parallel circuits. Both the combined capacitance equations look similar to the equations for combined resistance in series and parallel circuits. However, take note that they are the opposite way around to each other!

Capacitors in Series & Parallel (CIE A Level Physics)

Revision Note