Question

In the circuit of two cells, each of emf E but different internal resistances are connected in series as shown in the diagram. Determine the external resistance R in terms of internals resistances r₁ and r₂, such that potential drop across the cell E₁ is zero.

Answer 1

Current through the circuit 

I = 2E/(R+r₁+r₂)

Potential drop V₁ across the first cell, 

V₁ = E - Ir₁

Potential drop V₂ across the first cell, 

V₂ = E - Ir₂

For the potential drop V₁ across first cell to be zero, 

V₁ = E - Ir₁ = 0 

E = Ir₁ 

I = E/r₁ 

E/r₁ = 2E/(R+r₁+r₂)

Transposing and solving, 

R = r₁ - r₂ 

So if R = r₁ - r₂, the potential drop across the cell E₁ with internal resistance r₁ will be zero.