The wrong relation for modulus of rigidity (η) is

The correct option representing the wrong relation is η = Y/2(1-σ).

Let us analyze each of the given options step-by-step to understand why this relation is incorrect:

1. Definition of Modulus of Rigidity:
The modulus of rigidity ($\eta$) is defined as the ratio of shearing stress to shearing strain. Therefore, the relation:
$\eta =\frac{\text{Shearing stress}}{\text{Shearing strain}}$
is correct.

2. Unit of Modulus of Rigidity:
Since shearing strain is a dimensionless ratio (change in dimension over original dimension), the unit of the modulus of rigidity is the same as the unit of stress. The SI unit of stress is Newtons per square meter (${\text{N/m}}^2$). Thus, the statement "Unit of $\eta$ is ${\text{N/m}}^2$" is correct.

3. Relationship between Elastic Constants:
In theory of elasticity, the relation connecting Young's modulus ($Y$), modulus of rigidity ($\eta$), and Poisson's ratio ($\sigma$) is given by:
$Y=2\eta (1+\sigma )$
By rearranging this formula to solve for the modulus of rigidity ($\eta$), we get:
$\eta =\frac{Y}{2(1+\sigma )}$
This confirms that the relation $\eta =\frac{Y}{2(1+\sigma )}$ is correct.

Conclusion:
Comparing this with the third option, we see that:
$\eta =\frac{Y}{2(1-\sigma )}$
has a minus sign instead of a plus sign in the denominator. Therefore, this relation is incorrect (wrong).