An unknown resistance r si determined in terms of a standared resistance `R = 100 Omega` by using potentiometer. The potential difference across r is balanced againt `45 cm` length of the wire and that `(r + R)` is obtained at `70 cm` of the wire. Find the value of the unknown resistance.

Options

A. `200 Omega`
B. `280 Omega`
C. `180 Omega`
D. `100 Omega`

Explanation

To find the unknown resistance \( r \) using the potentiometer, we can use the relationship between the lengths of the wire and the resistances.

### Step-by-Step Solution:

1. **Identify the Given Values**:
- Standard resistance \( R = 100 \, \Omega \)
- Length of wire for \( r \): \( L_1 = 45 \, \text{cm} \)
- Length of wire for \( r + R \): \( L_2 = 70 \, \text{cm} \)

2. **Use the Potentiometer Formula**:
The formula relating the resistances and lengths is given by:
\[ \frac{r}{R} = \frac{L_1}{L_2} \]
Rearranging this gives:
\[ r = R \cdot \frac{L_1}{L_2} \]

3. **Substitute the Known Values**:
Substitute \( R = 100 \, \Omega \), \( L_1 = 45 \, \text{cm} \), and \( L_2 = 70 \, \text{cm} \) into the equation:
\[ r = 100 \cdot \frac{45}{70} \]

4. **Calculate the Unknown Resistance**:
First, calculate \( \frac{45}{70} \):
\[ \frac{45}{70} = \frac{9}{14} \approx 0.642857 \]
Now multiply by 100:
\[ r = 100 \cdot 0.642857 \approx 64.2857 \, \Omega \]

5. **Final Calculation**:
To get the exact value:
\[ r = \frac{100 \cdot 45}{70} = \frac{4500}{70} = 64.2857 \, \Omega \]

Thus, the value of the unknown resistance \( r \) is approximately \( 64.29 \, \Omega \).

### Final Answer:
The unknown resistance \( r \) is approximately \( 64.29 \, \Omega \).