19.4.2 Parallel Combination of Capacitors | Electric Potential And Capacitor | Physics 12

When capacitors are connected in parallel, both terminals of each capacitor are connected directly to the same two nodes of the circuit. This means every capacitor shares the same potential difference as the source.

Key Characteristics

Quantity	Behaviour in Parallel
Potential difference ( $V$ )	Same across each capacitor
Charge ( $Q$ )	Distributed — each capacitor stores a different charge
Equivalent capacitance ( $C_{e q}$ )	Sum of all individual capacitances

Derivation of Equivalent Capacitance

Consider three capacitors $C_{1}$ , $C_{2}$ , and $C_{3}$ connected in parallel across a voltage source $V$ .

Step 1 — Voltage is the same across each capacitor: $V_{1} = V_{2} = V_{3} = V$

Step 2 — Charge on each capacitor: $Q_{1} = C_{1} V, Q_{2} = C_{2} V, Q_{3} = C_{3} V$

Step 3 — Total charge drawn from the source: $Q = Q_{1} + Q_{2} + Q_{3}$ $Q = C_{1} V + C_{2} V + C_{3} V$ $Q = (C_{1} + C_{2} + C_{3}) V$

Step 4 — Define equivalent capacitance $C_{e q}$ such that $Q = C_{e q} V$ : $C_{e q} = C_{1} + C_{2} + C_{3}$

For $n$ capacitors in parallel: $C_{e q} = C_{1} + C_{2} + C_{3} + \dots + C_{n}$

Key result: The equivalent capacitance in a parallel combination is always greater than the largest individual capacitance.

Physical Interpretation

Connecting capacitors in parallel effectively increases the total plate area $A$ available for storing charge while keeping the plate separation $d$ constant. Since $C = ε_{0} A / d$ , a larger effective area means a larger capacitance.

Charge Distribution

Because $V$ is the same for all capacitors, the charge stored on each is proportional to its capacitance: $\frac{Q _{1}}{Q _{2}} = \frac{C _{1}}{C _{2}}$

A larger capacitor stores more charge at the same voltage.

Worked Example

Problem: Three capacitors of $2 μ F$ , $3 μ F$ , and $5 μ F$ are connected in parallel to a $10 V$ battery. Find (a) the equivalent capacitance, (b) the total charge, and (c) the charge on each capacitor.

Solution:

(a) Equivalent capacitance: $C_{e q} = 2 + 3 + 5 = 10 μ F$

(b) Total charge: $Q = C_{e q} V = 10 μ F \times 10 V = 100 μ C$

(c) Charge on each capacitor (all at $V = 10 V$ ): $Q_{1} = 2 μ F \times 10 V = 20 μ C$ $Q_{2} = 3 μ F \times 10 V = 30 μ C$ $Q_{3} = 5 μ F \times 10 V = 50 μ C$

Check: $20 + 30 + 50 = 100 μ C$ ✓

Comparison: Series vs Parallel

Property	Series	Parallel
Same quantity	Charge $Q$	Voltage $V$
$C_{e q}$ vs individual	Smaller than smallest	Larger than largest
Use case	High voltage distribution	Large charge storage

Quantity

Behaviour in Parallel

Potential difference (

V

)

Same across each capacitor

Charge (

Q

)

Distributed — each capacitor stores a different charge

Equivalent capacitance (

C_{e q}

)

Sum of all individual capacitances

Derivation of Equivalent Capacitance

Consider three capacitors

C_{1}

C_{2}

, and

C_{3}

connected in parallel across a voltage source

V

Step 1 — Voltage is the same across each capacitor:

V_{1} = V_{2} = V_{3} = V

Step 2 — Charge on each capacitor:

Q_{1} = C_{1} V, Q_{2} = C_{2} V, Q_{3} = C_{3} V

Step 3 — Total charge drawn from the source:

Q = Q_{1} + Q_{2} + Q_{3}

Q = C_{1} V + C_{2} V + C_{3} V

Q = (C_{1} + C_{2} + C_{3}) V

Step 4 — Define equivalent capacitance

C_{e q}

such that

Q = C_{e q} V

C_{e q} = C_{1} + C_{2} + C_{3}

For

n

capacitors in parallel:

C_{e q} = C_{1} + C_{2} + C_{3} + \dots + C_{n}

Key result: The equivalent capacitance in a parallel combination is always greater than the largest individual capacitance.

Worked Example

Problem: Three capacitors of

2 μ F

3 μ F

, and

5 μ F

are connected in parallel to a

10 V

battery. Find (a) the equivalent capacitance, (b) the total charge, and (c) the charge on each capacitor.

Solution:

(a) Equivalent capacitance:

C_{e q} = 2 + 3 + 5 = 10 μ F

(b) Total charge:

Q = C_{e q} V = 10 μ F \times 10 V = 100 μ C

V = 10 V

Q_{1} = 2 μ F \times 10 V = 20 μ C

Q_{2} = 3 μ F \times 10 V = 30 μ C

Q_{3} = 5 μ F \times 10 V = 50 μ C

Check:

20 + 30 + 50 = 100 μ C

✓

Property

Series

Parallel

Same quantity

Charge

Q

Voltage

V

C_{e q}

vs individual

Smaller than smallest

Larger than largest

Use case

High voltage distribution

Large charge storage