Current Electricity • NEET Revision

Example 3.34

EMF & Internal Resistance

Question

The reading on a high resistance voltmeter when a cell is connected across it is 3 V. When the terminals of the cell are also connected to a resistance of 4 Ω, the voltmeter reading drops to 1.2 V. Find the internal resistance of the cell.

Given

High-resistance voltmeter reading (open circuit) \(\Rightarrow\) EMF, \(E = 3\ \text{V}\)
External resistance \(R = 4\ \Omega\)
New voltmeter reading (terminal voltage) \(V = 1.2\ \text{V}\)

Asked

Internal resistance \(r\) of the cell.

Formula

\(V = E - Ir\) (terminal voltage with current flowing)

\(I = \dfrac{V}{R}\) (same current flows through \(R\))

\(r = \dfrac{(E - V)}{V}\,R\)

Solution

A high-resistance voltmeter draws negligible current, so its open-circuit reading equals the EMF: \(E = 3\ \text{V}\).
With \(R = 4\ \Omega\) connected, current in circuit: \(I = \dfrac{V}{R} = \dfrac{1.2}{4} = 0.3\ \text{A}\).
Voltage lost inside the cell: \(E - V = 3 - 1.2 = 1.8\ \text{V}\).
Internal resistance: \(r = \dfrac{E - V}{I} = \dfrac{1.8}{0.3} = 6\ \Omega\).

\(r = 6\ \Omega\)

Short Cut

One-line plug-in: \(r = R\left(\dfrac{E}{V} - 1\right) = 4\left(\dfrac{3}{1.2} - 1\right) = 4(2.5 - 1) = 6\ \Omega\). Memorise \(r = R\big(\tfrac{E}{V}-1\big)\) — it appears again and again in NEET.