A voltmeter resistance `500 Omega` is used to measure the emf of a cell of internal resistance `4 Omega`. The percentage error in the reading of the voltmeter will be

Options

A. `0.4%`
B. `0.6%`
C. `0.8%`
D. `1.2%`

Explanation

To find the percentage error in the reading of the voltmeter when measuring the emf of a cell with internal resistance, we can follow these steps:

### Step 1: Understand the Circuit
The circuit consists of a cell with an emf \( E \) and an internal resistance \( r = 4 \, \Omega \). The voltmeter has a resistance \( R_v = 500 \, \Omega \). When the voltmeter is connected across the terminals of the cell, it will affect the total resistance in the circuit.

**Hint:** Identify the components of the circuit and their resistances.

### Step 2: Calculate the Total Resistance
The total resistance in the circuit when the voltmeter is connected is the sum of the internal resistance of the cell and the resistance of the voltmeter:
\[ R_{total} = r + R_v = 4 \, \Omega + 500 \, \Omega = 504 \, \Omega \]

**Hint:** Remember that resistances in series add up.

### Step 3: Calculate the Current in the Circuit
Using Ohm's law, the current \( I \) flowing through the circuit can be calculated as:
\[ I = \frac{E}{R_{total}} = \frac{E}{504 \, \Omega} \]

**Hint:** Current is determined by the total resistance and the emf.

### Step 4: Calculate the Voltage Drop Across the Internal Resistance
The voltage drop across the internal resistance of the cell is given by:
\[ V_{drop} = I \cdot r = \left(\frac{E}{504}\right) \cdot 4 \]

**Hint:** Use the formula for voltage drop, which is current times resistance.

### Step 5: Calculate the Reading of the Voltmeter
The voltmeter reading \( V_m \) will be the emf minus the voltage drop across the internal resistance:
\[ V_m = E - V_{drop} = E - \left(\frac{E \cdot 4}{504}\right) = E \left(1 - \frac{4}{504}\right) = E \left(\frac{500}{504}\right) \]

**Hint:** The voltmeter reading is the actual emf minus the voltage drop due to internal resistance.

### Step 6: Calculate the Percentage Error
The percentage error in the reading of the voltmeter can be calculated as:
\[ \text{Percentage Error} = \frac{\Delta E}{E} \times 100 \]
Where \( \Delta E \) is the difference between the actual emf and the voltmeter reading:
\[ \Delta E = E - V_m = E - E \left(\frac{500}{504}\right) = E \left(1 - \frac{500}{504}\right) = E \left(\frac{4}{504}\right) \]
Thus,
\[ \text{Percentage Error} = \frac{E \left(\frac{4}{504}\right)}{E} \times 100 = \frac{4}{504} \times 100 \approx 0.7937\% \]

### Final Answer
The percentage error in the reading of the voltmeter is approximately \( 0.8\% \).

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