Combination of Cells in Series and Parallel

A combination of Cells is the arrangement of two or more cells (batteries) connected together in an electrical circuit to obtain the desired voltage and current required for the proper operation of devices. Since a single cell may not always provide sufficient energy, cells are combined in different ways to enhance the overall performance of the circuit.

This arrangement is classified into two types:

1. Series Combination

When cells are connected in series, the same current flows through each cell. The total emf depends on how the cells are connected, while internal resistances always add. Consider two cells with emfs ε₁,ε_2, and internal resistances r₁, r_2, and current I.

Potential difference across first cell:

Potential difference across second cell:

Total potential difference:

Hence,

If one cell is reversed:

So,

For n identical cells:

If m cells are reversed:

2. Parallel Combination

When two or more cells are connected such that all positive terminals are joined together and all negative terminals are joined together, it is called a parallel combination of cells. It increases current capacity while voltage remains almost the same.

Total current in circuit:

Potential difference using first cell:

Potential difference using second cell:

Current through each cell:

Total current expression:

Equivalent resistance (parallel):

Simplified equivalent resistance:

Relation for equivalent emf:

Equivalent emf formula

Equivalent resistance for n cells:

Equivalent emf for n cells:

For identical cells:

Solved Problems

Question 1: Batteries of 10V and 5 V are connected in series such that their emf's point in the same direction. Find the equivalent EMF for the system. 

Solution: The formula for equivalent series emf is given by,

E_eq = E₁ + E₂ + ... 

Given: E₁ = 10, E₂  = 5

Substituting these values in the equation, 

E = E₁ + E₂

⇒ E = 10 + 5  

⇒ E = 15 V

Question 2: Batteries of 10V and 5 V are connected in series such that their emf's point in the same direction. The internal resistances of the batteries are 2 and 10 ohms respectively. Find the equivalent resistance for the system. 

Solution: Equivalent resistance is also given by a similar equation, 

r_eq = r₁ + r₂

Given: r₁ = 2, r₂  = 10

substituting these values in the equation, 

r = r₁ + r₂

⇒ r = 2 + 10  

⇒ r = 12 ohms

Question 3: Three batteries of internal resistances 2, 2, and 4 ohms are connected in parallel. Find the equivalent resistance for the system. 

Solution: The formula for equivalent resistance is given by,

Given: R₁ = 2, R₂  = 2 and R₃ = 4 

Substituting these values in the equation, 

⇒ 

Substitute

⇒ 

⇒ 

Question 4: Three batteries of internal resistances 5, 5 ohm, and 10, 10 V are connected in parallel. Find the equivalent resistance and emf for the system. 

Solution: Given

r₁ = 5 Ω, r₂ = 5 Ω, r₃ = 10 Ω
E₁ = 10 V, E₂ = 10 V, E₃ = 10 V

Equivalent Resistance:

Equivalent EMF:

Unsolved Problems

Question 1: Two batteries of 12 V and 8 V with internal resistances 3 Ω and 2 Ω are connected in series such that their EMFs point in the same direction. Find the equivalent EMF and equivalent internal resistance of the system.

Question 2: Three resistors of 4 Ω, 6 Ω, and 12 Ω are connected in series. Find the equivalent resistance of the combination.

Question 3: Two batteries of EMFs 9 V and 6 V with internal resistances 1 Ω and 3 Ω are connected in parallel. Find the equivalent EMF and equivalent resistance of the system.

Question 4: Four resistors of 2 Ω, 4 Ω, 6 Ω, and 12 Ω are connected in parallel. Calculate the equivalent resistance of the combination.

Question 5: Three batteries of EMFs 5 V, 10 V, and 15 V with internal resistances 2 Ω, 3 Ω, and 5 Ω are connected in series but in opposite directions (the second battery is reversed). Find the equivalent EMF and equivalent resistance of the system.