A cylinder is of length l and diameter d. On stretching the cylinder, an increment `Deltal` in length and decrease `Deltad` in diameter are caused. The Poisson ratio is

Options

A. `sigma=-(Deltal)/lxxd/(Deltad)`
B. `sigma=-l/d ×(Deltad)/(Deltal)`
C. `sigma=-(Deltal)/lxx(Deltad)/d`
D. `sigma=-l/(Deltal)xxd/(Deltad)`

Answer

To find the Poisson's ratio for the given cylinder, we need to follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The Poisson's ratio (σ) is defined as the ratio of lateral strain to longitudinal strain.
- Lateral strain is the strain in the direction perpendicular to the applied force, while longitudinal strain is the strain in the direction of the applied force.

2. **Identify the Variables**:
- Let the original length of the cylinder be \( l \).
- Let the original diameter of the cylinder be \( d \).
- When the cylinder is stretched, the change in length is \( \Delta l \) and the change in diameter is \( \Delta d \).

3. **Calculate Longitudinal Strain**:
- Longitudinal strain is given by:
\[ \text{Longitudinal Strain} = \frac{\Delta l}{l} \]

4. **Calculate Lateral Strain**:
- Lateral strain is given by:
\[ \text{Lateral Strain} = \frac{\Delta d}{d} \]
- Since the diameter decreases when the cylinder is stretched, we consider this change as negative:
\[ \text{Lateral Strain} = -\frac{\Delta d}{d} \]

5. **Substitute into the Poisson's Ratio Formula**:
- The Poisson's ratio is given by:
\[ \sigma = \frac{\text{Lateral Strain}}{\text{Longitudinal Strain}} = \frac{-\frac{\Delta d}{d}}{\frac{\Delta l}{l}} \]

6. **Simplify the Expression**:
- Rearranging the above expression gives:
\[ \sigma = -\frac{\Delta d \cdot l}{\Delta l \cdot d} \]

### Final Expression:
Thus, the Poisson's ratio for the material of the cylinder is:
\[ \sigma = -\frac{l}{d} \cdot \frac{\Delta d}{\Delta l} \]