A wire of length L and area of cross-section A, is stretched by a load. The elongation produced in the wire is I. If Y is the Young's modulus of the material of the wire, then the force constant of the wire is

Options

A. `(YL)/(A)`
B. `(Yl)/(A)`
C. `(YA)/(L)`
D. `(YA)/(l)`

Explanation

To find the force constant \( k \) of a wire given its Young's modulus \( Y \), length \( L \), area of cross-section \( A \), and elongation \( l \), we can follow these steps:

### 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:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
where stress is defined as \( \frac{F}{A} \) (force per unit area) and strain is defined as \( \frac{\Delta L}{L} \) (change in length per original length).

2. **Substitute Definitions**:
Using the definitions of stress and strain, we can rewrite Young's modulus as:
\[ Y = \frac{F/A}{\Delta L/L} \]
Rearranging gives:
\[ Y = \frac{F \cdot L}{A \cdot \Delta L} \]

3. **Rearranging for Force**:
From the equation above, we can express the force \( F \) in terms of Young's modulus \( Y \), the area \( A \), the original length \( L \), and the elongation \( \Delta L \):
\[ F = Y \cdot \frac{A \cdot \Delta L}{L} \]

4. **Using Hooke's Law**:
According to Hooke's Law, the force \( F \) can also be expressed in terms of the force constant \( k \) and the elongation \( l \):
\[ F = k \cdot l \]

5. **Equating the Two Expressions for Force**:
Now, we can set the two expressions for force equal to each other:
\[ k \cdot l = Y \cdot \frac{A \cdot l}{L} \]

6. **Solving for the Force Constant \( k \)**:
To find \( k \), we can divide both sides by \( l \) (assuming \( l \neq 0 \)):
\[ k = Y \cdot \frac{A}{L} \]

### Final Result:
Thus, the force constant \( k \) of the wire is given by:
\[ k = \frac{Y \cdot A}{L} \]