Question

Derive the relation for the internal resistance of a cell.

Answer 1

      Internal resistance of a cell

Let a cell of emf ε and internal resistance r be connected to an external resistance R as shown in the figure.

The total resistance of the circuit containing an external resistance R and internal resistance r is given by R + r.

According to Ohm's Law, the total current flowing through this circuit is expressed as:

\(I=\frac{V}{R}=\frac{\varepsilon }{R+r}\)

By rearranging this equation, the total electromotive force ε of the cell can be written as:

ε = I(R + r) = IR + Ir

Since the terminal voltage is V = IR and the internal voltage drop is v = Ir, substituting these values gives the standard relation:

ε = V + v

or v = ε − V

Finally, by dividing the internal voltage drop by the current and substituting \(I = \frac{V}{R}\), the relation for internal resistance is derived as:

r = `v/I`

= `(ε - V)/(V/R)`

= `R(ε/V - 1)`