The area of cross section of a steel wire `(Y=2.0 xx 10^(11) N//m^(2))` is `0.1 cm^(2)`. The force required to double is length will be

Options

A. `2xx10^(12) N`
B. `2xx10^(11) N`
C. `2xx10^(10) N`
D. `2xx10^(6) N`

Answer

To solve the problem, we need to find the force required to double the length of a steel wire given its Young's modulus and cross-sectional area. We will follow these steps:

### Step 1: Convert the area of cross-section from cm² to m²
The area of cross-section is given as \(0.1 \, \text{cm}^2\). We need to convert this into square meters.

\[ 0.1 \, \text{cm}^2 = 0.1 \times 10^{-4} \, \text{m}^2 = 1.0 \times 10^{-5} \, \text{m}^2 \]

### Step 2: Define the initial and final lengths of the wire
Let the initial length of the wire be \(L\). If the length is doubled, the final length will be:

\[ \text{Final Length} = 2L \]

### Step 3: Calculate the strain in the wire
Strain is defined as the change in length divided by the original length. Thus, the strain can be calculated as follows:

\[ \text{Strain} = \frac{\text{Final Length} - \text{Initial Length}}{\text{Initial Length}} = \frac{2L - L}{L} = \frac{L}{L} = 1 \]

### Step 4: Use Young's modulus to find the stress
Young's modulus \(Y\) is defined as the ratio of stress to strain:

\[ Y = \frac{\text{Stress}}{\text{Strain}} \]

Given that the strain is 1, we can express stress as:

\[ \text{Stress} = Y \times \text{Strain} = Y \times 1 = Y \]

### Step 5: Relate stress to force and area
Stress is also defined as force per unit area:

\[ \text{Stress} = \frac{F}{A} \]

Where \(F\) is the force applied and \(A\) is the area of cross-section. Therefore, we can set the two expressions for stress equal to each other:

\[ \frac{F}{A} = Y \]

### Step 6: Solve for the force \(F\)
Rearranging the equation gives us:

\[ F = Y \times A \]

Substituting the values for \(Y\) and \(A\):

\[ F = (2.0 \times 10^{11} \, \text{N/m}^2) \times (1.0 \times 10^{-5} \, \text{m}^2) \]

\[ F = 2.0 \times 10^{6} \, \text{N} \]

### Final Answer
The force required to double the length of the steel wire is:

\[ \boxed{2.0 \times 10^{6} \, \text{N}} \]