Question

If the longitudinal strain for a wire is $0.003$ and its Poisson’s ratio is $0.5$, then its lateral strain isA) $0.003$B) $0.0075$C) $0.015$D) $0.4$

Hint: For a wire, longitudinal strain refers to the ratio of change in length to the original length of the wire when the wire is stretched along the length. Lateral strain refers to the ratio of change in diameter to the original diameter of wire due to the force applied along the length of the wire. Poisson’s ratio gives a relation between the former and the latter.

Poisson’s ratio is defined as the ratio of the lateral strain to the longitudinal strain. It is represented by $\nu$ and is given by

$\nu =\dfrac{lateral\_strain}{longitudinal\_strain}$

It has no unit. It is seen that most of the materials have a Poisson’s ratio ranging between $0.0$and $0.5$. Poisson’s ratio is a measure of Poisson’s effect, which is the phenomenon in which a material tends to expand in a direction perpendicular to the compression or vice versa.

For a wire, Poisson’s ratio can be deduced from the values of longitudinal strain and lateral strain. When a wire is stretched, there is change in both length and diameter of the wire. The length tends to increase while the diameter tends to decrease. Longitudinal strain is the ratio of the change in length to the original length of the wire. Lateral strain is the ratio of the change in diameter to the original diameter of the wire. Longitudinal strain is also called as axial strain because the change in length is along the direction of applied force while lateral strain is also called as transverse strain because the change in diameter is perpendicular to the direction of applied force. Both longitudinal strain and lateral strain have no units and are given by $\dfrac{\Delta l}{l}$and $\dfrac{\Delta d}{d}$, respectively.

Therefore, Poisson’s ratio for a wire is given by
$\nu =\dfrac{lateral\_strain}{longitudinal\_strain}=\dfrac{\dfrac{\Delta l}{l}}{\dfrac{\Delta d}{d}}$
In the question provided, we have Poisson’s ratio equal to $0.5$ and longitudinal strain equal to $0.003$. It is clear that later strain is equal to
$lateral\_strain=\nu \times longitudinal\_strain=0.5\times 0.003=0.015$

So, the correct answer is “Option C”.

Note:
In the above explanation, only the magnitudes of parameters are considered. If directions are considered, the sign can change from positive to negative. For example, when a rubber cube is stretched, the length decreases while the width increases. Longitudinal strain is negative while the lateral strain is positive. Poisson’s ratio turns out to be negative. To avoid this confusion, Poisson’s ratio is usually defined as the negative of the ratio of the lateral strain to the longitudinal strain.