If the ammeter A shows a zero reading in the circuit shown below, the value of resistance R is
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In null-deflection or bridge-balancing problems, treat the branch with zero current as an open circuit to simplify the calculation of potential at the balancing nodes.
Concept:
A "zero reading" in an ammeter indicates a null point, meaning no current flows through that specific branch. This occurs when the potential at the junction equals the opposing battery's EMF.
• Null Condition: The potential difference across the ammeter branch is zero.
• Potential Divider: The potential at the junction between the resistors is determined by their relative values in the main loop.
Step 1: Analyze the condition for zero ammeter current.
For the ammeter to read zero, the potential drop across the unknown resistor $R$ in the main circuit must be exactly equal to the EMF of the balancing battery, which is $2\text{ V}$.
When these potentials are equal, no current flows from the $2\text{ V}$ source into the junction.
Step 2: Apply the potential divider rule to the primary loop.
Since no current enters the ammeter branch, the total current from the $10\text{ V}$ battery flows only through the $500\ \Omega$ resistor and the resistor $R$ in series.
The potential across $R$ is given by:
\[ V_R = \frac{R}{500 + R} \times 10\text{ V} \]
Step 3: Solve for the resistance R.
Set the potential across $R$ equal to the $2\text{ V}$ balancing voltage:
\[ 2 = \frac{10R}{500 + R} \]
Multiply both sides by $(500 + R)$:
\[ 1000 + 2R = 10R \]
Subtract $2R$ from both sides:
\[ 1000 = 8R \]
\[ R = \frac{1000}{8} = 125\ \Omega \]
