Question

Three capacitors C₁ , C₂ and C₃ are connected to a cell of emf V as shown in the figure.

1. The arrangement of these three capacitors are called…………

• parallel combination
• series combination
• LCR combination
• c-c combination

2. Find the effective capacitance of the above combination.

The above graph shows the variation of potential in going from a to g. From the graph the relation among C₁ , C₂ and C₃ is

• C₁ = C₂ = C₃
• 2C₁ = 2C₂ = C₃
• C₁ = C₂ = 2C₃
• C₁ = C₃ = C₃

Answer 1

1. Series connection.

2. Capacitors in series:

Let three capacitors C₁ ,C₂ and C₃ be connected in series to p.d of V. Let V₁ , V₂ and V₃ be the voltage across C₁ , C₂ and C₃.

The applied voltage can be written as,

V = V₁ + V₂ + V₃ …..(1)

Charge ‘q’ is same as in all the capacitor. So,

V₁ = \(\frac{q}{C_1}\) , V₂ = \(\frac{q}{C_2}\) and V₃ = \(\frac{q}{C_3}\)

Substituting these values in (1),

V = \(\frac{q}{C_1}+\frac{q}{C_2}+\frac{q}{C_3}\) .......(2)

If these capacitors are replaced by a equivalent capacitance ‘C’, then

V = \(\frac{q}{C}\)

Hence eq(2) can be written as

\(\frac{q}{C}=\frac{q}{C_1}+\frac{q}{C_2}+\frac{q}{C_3}\)

\(\frac{1}{C}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}\)

Effective capacitance is decreased by series combination.

3. From the graph, we get v₁ = 1 v, v₂ = 1 v, v₃ = 2v.

This arrangement is series. Hence charge stored in each capacitor is same.

∴ C₁ = \(\frac{Q}{1},C_2=\frac{Q}{1}\)

C₃ = \(\frac{Q}{2}\)

C₁ : C₂ : C₃ 

1: 1: \(\frac{1}{2}\)

2: 2: 1 

2C₁ : 2C₂ : C₃