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Step-by-Step Solution
Step 1: Understand the Setup
We have two identical cells, each with emf 1.5\,\text{V} , connected in series across a 10\,\Omega resistor. An ideal voltmeter placed across this 10\,\Omega resistor reads 1.5\,\text{V} . We want to find the internal resistance r of each cell.
Step 2: Determine the Total EMF and Total Internal Resistance
Since the cells are in series, their emfs add up. Thus, the total emf of the two cells is:
E_{\text{total}} = 1.5 \,\text{V} + 1.5\,\text{V} = 3\,\text{V}.
If each cell has internal resistance r , then the total internal resistance of the two cells in series is
r_{\text{total}} = r + r = 2r.
Step 3: Current Through the Circuit
The ideal voltmeter reads 1.5\,\text{V} across the 10\,\Omega resistor. By Ohm’s law, the current I through the resistor is
I = \frac{V_{\text{across resistor}}}{R} = \frac{1.5\,\text{V}}{10\,\Omega} = 0.15\,\text{A}.
This current flows through the entire series circuit (the cells and the external resistor).
Step 4: Apply Kirchhoff’s Voltage Law
Kirchhoff’s Voltage Law states that the total emf in the circuit equals the sum of all voltage drops:
E_{\text{total}} = I \left( R + r_{\text{total}} \right).
Substituting the values, we have
3\,\text{V} = 0.15\,\text{A} \times \left( 10\,\Omega + 2r \right).
Step 5: Solve for the Internal Resistance r
Expand and solve for r :
3\,\text{V} = 0.15\,\text{A} \times 10\,\Omega + 0.15\,\text{A} \times 2r
3\,\text{V} = 1.5\,\text{V} + 0.3\,\text{A}\,r
3\,\text{V} - 1.5\,\text{V} = 0.3\,\text{A}\,r
1.5\,\text{V} = 0.3\,\text{A}\,r
r = \frac{1.5\,\text{V}}{0.3\,\text{A}} = 5\,\Omega.
Final Answer
The internal resistance of each cell is 5\,\Omega .
