Question

In the given circuit, if the potential difference between the points A and B is 90 V and the resistance of the voltmeter is 6000 $\Omega$, then the reading of the voltmeter is

• A. 40 V
• B. 50 V
• C. 30 V
• D. 45 V

Hint:

Whenever a voltmeter has finite resistance, do not treat it as an ideal voltmeter. First combine the voltmeter resistance with the resistor across which it is connected using the parallel combination formula: \[ R_p=\frac{R_1R_2}{R_1+R_2} \] and then apply Ohm's law or voltage division.

Answer 1

The Correct Option is B

Concept: The voltmeter is connected across the \(2000\Omega\) resistor. Since the voltmeter has finite resistance, it is connected in parallel with the \(2000\Omega\) resistor. Therefore, we first find the equivalent resistance of the parallel combination and then use the voltage division rule.

Step 1: Calculate the equivalent resistance of the parallel combination.
The voltmeter resistance is \[ R_V=6000\Omega \] The resistor across which it is connected is \[ R=2000\Omega \] Equivalent resistance: \[ R_p=\frac{R\times R_V}{R+R_V} \] \[ R_p=\frac{2000\times6000}{2000+6000} \] \[ R_p=\frac{12000000}{8000} \] \[ R_p=1500\Omega \] Thus the parallel combination is equivalent to \[ 1500\Omega \]

Step 2: Find the total resistance of the circuit.
The \(1200\Omega\) resistor is in series with the parallel combination. Hence, \[ R_{\text{total}} = 1200+1500 \] \[ R_{\text{total}} = 2700\Omega \]

Step 3: Calculate the current flowing through the circuit.
Given, \[ V_{AB}=90V \] Using Ohm's law, \[ I=\frac{V}{R} \] \[ I=\frac{90}{2700} \] \[ I=\frac{1}{30}A \] \[ I=0.0333A \]

Step 4: Find the voltage across the parallel combination.
The voltmeter measures the voltage across the parallel branch. Therefore, \[ V_V=I\times R_p \] \[ V_V=\frac{1}{30}\times1500 \] \[ V_V=50V \]

Step 5: State the answer.
Hence the voltmeter reading is \[ \boxed{50V} \] Therefore, \[ \boxed{\text{Option (B)}} \]