The young's modulus of a wire of length (L) and radius (r ) is Y. If the length is reduced to` L/2` and radius `r/2` , then its young's modulus will be

Options

A. `(Y)/(2)`
B. Y
C. 2Y
D. 4Y

Explanation

To solve the problem, we need to understand the relationship between Young's modulus and the dimensions of the wire. Let's go through the solution step by step.

### Step 1: Understand Young's Modulus
Young's modulus (Y) is defined as the ratio of stress to strain in a material. It is given by the formula:
\[ Y = \frac{\text{Stress}}{\text{Strain}} \]
where:
- Stress = \(\frac{F}{A}\) (Force per unit area)
- Strain = \(\frac{\Delta L}{L}\) (Change in length per original length)

### Step 2: Analyze the Initial Conditions
Initially, the wire has:
- Length = \(L\)
- Radius = \(r\)

The cross-sectional area \(A\) of the wire can be calculated using the formula for the area of a circle:
\[ A = \pi r^2 \]
Thus, the initial stress (\(\sigma\)) can be expressed as:
\[ \sigma = \frac{F}{\pi r^2} \]

### Step 3: Analyze the New Conditions
Now, the length of the wire is reduced to \(L/2\) and the radius is reduced to \(r/2\).

The new cross-sectional area \(A'\) becomes:
\[ A' = \pi \left(\frac{r}{2}\right)^2 = \pi \frac{r^2}{4} = \frac{\pi r^2}{4} \]

The new stress (\(\sigma'\)) is:
\[ \sigma' = \frac{F}{A'} = \frac{F}{\frac{\pi r^2}{4}} = \frac{4F}{\pi r^2} \]

### Step 4: Calculate the New Strain
The new strain (\(\epsilon'\)) is calculated as:
\[ \epsilon' = \frac{\Delta L'}{L'} = \frac{L/2}{L/2} = 1 \]

### Step 5: Determine Young's Modulus in New Conditions
Now we can find the new Young's modulus \(Y'\):
\[ Y' = \frac{\sigma'}{\epsilon'} = \frac{4F/\pi r^2}{1} = \frac{4F}{\pi r^2} \]

### Step 6: Compare Young's Modulus
Notice that the new Young's modulus \(Y'\) is still dependent on the material properties and does not change with the dimensions of the wire. Thus:
\[ Y' = Y \]

### Conclusion
The Young's modulus remains the same regardless of the changes in length and radius. Therefore, the answer is:
\[ \text{Young's modulus after changes} = Y \]

### Final Answer
The Young's modulus will remain \(Y\).

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