A voltmeter of resistance `998 Omega` is connected across a cell of emf 2 V and internal resistance `2 Omega`. Find the p.d. across the voltmeter, that across the terminals of the cell and percentage error in the reading of the voltmeter.

Options

A. `1.99V`
B. `3.5 V`
C. 5 V
D. 6 V

Answer

To solve the problem, we will follow these steps:

### Step 1: Identify the given values
- EMF of the cell, \( e = 2 \, \text{V} \)
- Internal resistance of the cell, \( r = 2 \, \Omega \)
- Resistance of the voltmeter, \( R = 998 \, \Omega \)

### Step 2: Calculate the potential difference (p.d.) across the voltmeter
The formula to calculate the potential difference across the voltmeter when connected to a cell is given by:

\[ V = \frac{e \cdot R}{r + R} \]

Substituting the known values into the formula:

\[ V = \frac{2 \cdot 998}{2 + 998} \]

Calculating the denominator:

\[ r + R = 2 + 998 = 1000 \, \Omega \]

Now substituting back into the equation for \( V \):

\[ V = \frac{1996}{1000} = 1.996 \, \text{V} \]

### Step 3: Calculate the potential difference across the terminals of the cell
The potential difference across the terminals of the cell is equal to the EMF minus the voltage drop across the internal resistance. The voltage drop across the internal resistance can be calculated using Ohm's law:

\[ \text{Voltage drop} = I \cdot r \]

Where \( I \) is the current flowing through the circuit. The current \( I \) can be calculated as:

\[ I = \frac{e}{r + R} = \frac{2}{1000} = 0.002 \, \text{A} \]

Now, substituting this value back into the voltage drop equation:

\[ \text{Voltage drop} = 0.002 \cdot 2 = 0.004 \, \text{V} \]

Thus, the potential difference across the terminals of the cell is:

\[ \text{p.d. across terminals} = e - \text{Voltage drop} = 2 - 0.004 = 1.996 \, \text{V} \]

### Step 4: Calculate the percentage error in the reading of the voltmeter
The percentage error can be calculated using the formula:

\[ \text{Percentage Error} = \frac{\text{Actual Value} - \text{Measured Value}}{\text{Actual Value}} \times 100 \]

Where:
- Actual Value = p.d. across terminals = 1.996 V
- Measured Value = p.d. across voltmeter = 1.996 V

Substituting these values into the formula:

\[ \text{Percentage Error} = \frac{1.996 - 1.996}{1.996} \times 100 = 0\% \]

### Final Results
- Potential difference across the voltmeter: \( V = 1.996 \, \text{V} \)
- Potential difference across the terminals of the cell: \( 1.996 \, \text{V} \)
- Percentage error in the reading of the voltmeter: \( 0\% \)