The Young’s modulus of the material of a wire is 6 x 10¹² N / m² and there is no transverse strain in it, then its modulus of rigidity will be

The correct option is 3 x 10¹² N / m².

Step-by-step Explanation:
The relation between Young's modulus ($Y$), modulus of rigidity ($\eta$), and Poisson's ratio ($\sigma$) of a material is given by the formula:
$Y=2\eta (1+\sigma )$

Poisson's ratio is defined as the ratio of lateral (transverse) strain to longitudinal strain:
$\sigma =-\frac{\text{Transverse strain}}{\text{Longitudinal strain}}$

According to the problem, there is no transverse strain in the wire, which means:
$\text{Transverse strain}=0$
Therefore, the Poisson's ratio is:
$\sigma =0$

Substituting $\sigma =0$ into the relation between elastic constants:
$Y=2\eta (1+0)$
$Y=2\eta$

Solving for the modulus of rigidity ($\eta$):
$\eta =\frac{Y}{2}$

Given that the Young's modulus of the material is:
$Y=6\times {10}^{12}{\text{N/m}}^2$

Substitute the value of $Y$ into the equation:
$\eta =\frac{6\times {10}^{12}}{2}=3\times {10}^{12}{\text{N/m}}^2$