Two resistance of `400 Omega` and `800 Omega` are connected in series with 6 volt battery of negligible internal resistance. A voltmeter of resistance `10,000 Omega` is used to measure the potential difference across `400 Omega`. The error in measurement of potential difference in volts approximatley is

Options

A. 0.01
B. 0.02
C. 0.03
D. 0.05

Explanation

To solve the problem step-by-step, we will first find the actual potential difference across the 400 Ω resistor and then calculate the measured potential difference when the voltmeter is connected. Finally, we will find the error in the measurement.

### Step 1: Calculate the Total Resistance in the Circuit
The two resistors, 400 Ω and 800 Ω, are connected in series. Therefore, the total resistance \( R_{total} \) is given by:
\[ R_{total} = R_1 + R_2 = 400 \, \Omega + 800 \, \Omega = 1200 \, \Omega \]

**Hint:** Remember that in a series circuit, the total resistance is simply the sum of the individual resistances.

### Step 2: Calculate the Current in the Circuit
Using Ohm's Law, the current \( I \) flowing through the circuit can be calculated using the total voltage \( V \) and the total resistance \( R_{total} \):
\[ I = \frac{V}{R_{total}} = \frac{6 \, V}{1200 \, \Omega} = 0.005 \, A \, (or \, 5 \, mA) \]

**Hint:** Ohm's Law states that \( V = I \times R \). Rearranging it gives \( I = \frac{V}{R} \).

### Step 3: Calculate the Actual Potential Difference Across the 400 Ω Resistor
The potential difference \( V_{400} \) across the 400 Ω resistor can be calculated using Ohm's Law:
\[ V_{400} = I \times R_1 = 0.005 \, A \times 400 \, \Omega = 2 \, V \]

**Hint:** The voltage across a resistor in a series circuit can be found by multiplying the current through it by its resistance.

### Step 4: Calculate the Equivalent Resistance with the Voltmeter
When the voltmeter (10,000 Ω) is connected across the 400 Ω resistor, it forms a parallel combination with the 400 Ω resistor. The equivalent resistance \( R_{eq} \) of the 400 Ω resistor and the voltmeter is given by:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_{voltmeter}} = \frac{1}{400} + \frac{1}{10000} \]
Calculating this gives:
\[ \frac{1}{R_{eq}} = \frac{25 + 1}{10000} = \frac{26}{10000} \implies R_{eq} = \frac{10000}{26} \approx 384.6 \, \Omega \]

**Hint:** For resistors in parallel, the reciprocal of the total resistance is the sum of the reciprocals of the individual resistances.

### Step 5: Calculate the New Total Resistance in the Circuit
Now, the total resistance in the circuit when the voltmeter is connected is:
\[ R_{total\_new} = R_{eq} + R_2 = 384.6 \, \Omega + 800 \, \Omega = 1184.6 \, \Omega \]

**Hint:** When additional components are added in series, their resistances are added to the total resistance.

### Step 6: Calculate the Current with the Voltmeter Connected
The current \( I_{new} \) in the circuit with the voltmeter connected is:
\[ I_{new} = \frac{V}{R_{total\_new}} = \frac{6 \, V}{1184.6 \, \Omega} \approx 0.00506 \, A \]

**Hint:** Use the same formula for current as before, but with the new total resistance.

### Step 7: Calculate the Measured Potential Difference Across the 400 Ω Resistor
Now, we can calculate the potential difference \( V_{measured} \) across the equivalent resistance:
\[ V_{measured} = I_{new} \times R_{eq} = 0.00506 \, A \times 384.6 \, \Omega \approx 1.95 \, V \]

**Hint:** The measured voltage is calculated using the current through the equivalent resistance.

### Step 8: Calculate the Error in Measurement
The error in measurement can be calculated as:
\[ \text{Error} = V_{actual} - V_{measured} = 2 \, V - 1.95 \, V = 0.05 \, V \]

**Hint:** The error is simply the difference between the actual voltage and the measured voltage.

### Conclusion
The error in the measurement of potential difference across the 400 Ω resistor is approximately **0.05 volts**.