A potentiometer wire 10 long has a resistance of `40Omega`. It is connected in series with a resistances box and a 2 v storage cell. If the potential gradient along the wire is `0.01(V)/(m)` the resistance unplugged in the box is

Options

A. `260 Omega`
B. `760 Omega`
C. `960 Omega`
D. `1060 Omega`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Understand the given data
- Length of the potentiometer wire, \( L = 10 \, \text{m} \)
- Resistance of the potentiometer wire, \( R_{wire} = 40 \, \Omega \)
- Voltage of the storage cell, \( V = 2 \, \text{V} \)
- Potential gradient, \( k = 0.01 \, \text{V/m} \)

### Step 2: Calculate the potential difference across the potentiometer wire
The potential difference (\( V_{AB} \)) across the length of the potentiometer wire can be calculated using the formula:
\[ V_{AB} = k \times L \]
Substituting the values:
\[ V_{AB} = 0.01 \, \text{V/m} \times 10 \, \text{m} = 0.1 \, \text{V} \]

### Step 3: Calculate the current through the potentiometer wire
Using Ohm's law, the current (\( I \)) through the potentiometer wire can be calculated as:
\[ I = \frac{V_{AB}}{R_{wire}} \]
Substituting the values:
\[ I = \frac{0.1 \, \text{V}}{40 \, \Omega} = \frac{1}{400} \, \text{A} \]

### Step 4: Set up the equation for the total resistance in the circuit
The total resistance in the circuit (\( R_{total} \)) is the sum of the resistance of the potentiometer wire and the resistance in the box (\( R \)):
\[ R_{total} = R_{wire} + R = 40 \, \Omega + R \]

### Step 5: Use the total voltage and current to find the total resistance
According to Ohm's law, the total voltage supplied by the cell is equal to the product of the current and the total resistance:
\[ V = I \times R_{total} \]
Substituting the known values:
\[ 2 \, \text{V} = \left(\frac{1}{400} \, \text{A}\right) \times (40 + R) \]

### Step 6: Solve for \( R \)
Cross-multiplying gives:
\[ 2 \times 400 = 40 + R \]
\[ 800 = 40 + R \]
Now, isolate \( R \):
\[ R = 800 - 40 = 760 \, \Omega \]

### Conclusion
The resistance unplugged in the box is \( R = 760 \, \Omega \).

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