The Poisson's ratio of a material is 0.5. If a force is applied to a wire of this material, there is a decrease in the cross-sectional area by `4%`. The percentage increase in the length is

Options

A. `4%`
B. `2%`
C. `2.5%`
D. `4%`

Answer

To solve the problem step by step, we will use the definition of Poisson's ratio and the relationship between the change in dimensions of the wire.

### Step 1: Understand Poisson's Ratio
Poisson's ratio (ν) is defined as the ratio of the transverse strain to the longitudinal strain. Mathematically, it is given by:
\[ \nu = -\frac{\Delta r / r}{\Delta l / l} \]
where:
- \(\Delta r\) is the change in radius,
- \(r\) is the original radius,
- \(\Delta l\) is the change in length,
- \(l\) is the original length.

### Step 2: Relate Change in Area to Change in Radius
The area \(A\) of the wire is given by:
\[ A = \pi r^2 \]
The change in area \(\Delta A\) can be expressed as:
\[ \frac{\Delta A}{A} = 2 \frac{\Delta r}{r} \]
Given that the cross-sectional area decreases by 4%, we have:
\[ \frac{\Delta A}{A} = -0.04 \]
Thus, we can set up the equation:
\[ -0.04 = 2 \frac{\Delta r}{r} \]

### Step 3: Solve for Change in Radius
From the equation above, we can solve for \(\frac{\Delta r}{r}\):
\[ \frac{\Delta r}{r} = -0.02 \]
This indicates that the radius decreases by 2%.

### Step 4: Relate Change in Length to Change in Radius
Using Poisson's ratio, we can relate the change in length to the change in radius:
\[ \nu = -\frac{\Delta r / r}{\Delta l / l} \]
Substituting the known values:
\[ 0.5 = -\frac{-0.02}{\Delta l / l} \]
This simplifies to:
\[ 0.5 = \frac{0.02}{\Delta l / l} \]

### Step 5: Solve for Change in Length
Rearranging the equation gives:
\[ \Delta l / l = \frac{0.02}{0.5} = 0.04 \]
Thus, the percentage increase in length is:
\[ \Delta l / l = 0.04 = 4\% \]

### Final Answer
The percentage increase in the length of the wire is **4%**.

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