According to Hooke's law, the force required to change the length of a wire by `'l'` is proportional to

Options

A. `1//l^(2)`
B. `1//l`
C. `l`
D. `l^(2)`

Answer

To solve the question based on Hooke's Law, we need to establish the relationship between the force applied to a wire and the change in its length. Let's break it down step by step:

### Step 1: Understand Hooke's Law
According to Hooke's Law, the force (F) required to change the length of a wire is directly proportional to the change in length (ΔL). Mathematically, this can be expressed as:
\[ F \propto \Delta L \]

### Step 2: Define the Change in Length
In this case, we are given that the change in length is represented by 'l'. Therefore, we can write:
\[ F \propto l \]

### Step 3: Introduce Young's Modulus
Young's modulus (Y) relates stress and strain in a material. Stress is defined as force per unit area (F/A), and strain is defined as the change in length per original length (ΔL/L). Thus, we can express Young's modulus as:
\[ Y = \frac{\text{Stress}}{\text{Strain}} = \frac{F/A}{\Delta L/L} \]

### Step 4: Rearranging Young's Modulus
From the definition of Young's modulus, we can rearrange the equation to express force (F):
\[ F = Y \cdot A \cdot \frac{\Delta L}{L} \]

### Step 5: Analyze the Proportionality
In the equation \( F = Y \cdot A \cdot \frac{\Delta L}{L} \):
- Young's modulus (Y) is a constant for a given material.
- The cross-sectional area (A) is constant for a given wire.
- The original length (L) is also constant.

This indicates that the force (F) is directly proportional to the change in length (ΔL). Since we have defined ΔL as 'l', we can conclude:
\[ F \propto l \]

### Conclusion
Thus, according to Hooke's Law, the force required to change the length of a wire by 'l' is proportional to 'l'.

### Final Answer
The force required to change the length of a wire by 'l' is proportional to 'l'.

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