The Young's modulus of brass and steel are respectively `10 xx 10^(10) N//m^(2)`. And `2 xx 10^(10) N//m^(2)` A brass wire and a steel wire of the same length are extended by 1 mm under the same force, the radii of brass and steel wires ar `R_(B) and R_(S)`respectively. Then

Options

A. `R_(S ) = sqrt(2) R_(B)`
B. `R_(S) = R_(B)//sqrt(2)`
C. `R_(S) = 4 R_(B)`
D. `R_(S) = R_(B)//4`

Explanation

To solve the problem, we will use the definition of Young's modulus and the relationship between stress, strain, and the dimensions of the wires.

### Step-by-Step Solution:

1. **Understand Young's Modulus**:
Young's modulus (Y) is defined as the ratio of stress to strain. Mathematically, it is given by:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]
where \( F \) is the applied force, \( A \) is the cross-sectional area, \( \Delta L \) is the change in length, and \( L \) is the original length.

2. **Write the Expression for Each Wire**:
For the brass wire:
\[ Y_B = \frac{F}{\pi R_B^2} \cdot \frac{L}{\Delta L} \]
For the steel wire:
\[ Y_S = \frac{F}{\pi R_S^2} \cdot \frac{L}{\Delta L} \]

3. **Set Up the Ratio of Young's Modulus**:
Since both wires are subjected to the same force and have the same original length and change in length, we can set up the ratio:
\[ \frac{Y_B}{Y_S} = \frac{R_S^2}{R_B^2} \]

4. **Substitute the Values of Young's Modulus**:
Given:
- \( Y_B = 10 \times 10^{10} \, \text{N/m}^2 \)
- \( Y_S = 2 \times 10^{10} \, \text{N/m}^2 \)

Substitute these values into the ratio:
\[ \frac{10 \times 10^{10}}{2 \times 10^{10}} = \frac{R_S^2}{R_B^2} \]

5. **Simplify the Ratio**:
Simplifying the left side:
\[ \frac{10}{2} = 5 \implies \frac{R_S^2}{R_B^2} = 5 \]

6. **Find the Relationship Between Radii**:
Taking the square root of both sides gives:
\[ \frac{R_S}{R_B} = \sqrt{5} \]
Therefore, we can express \( R_S \) in terms of \( R_B \):
\[ R_S = R_B \cdot \sqrt{5} \]

### Final Answer:
The relationship between the radii of the brass and steel wires is:
\[ R_S = R_B \cdot \sqrt{5} \]