Choose the correct relationship between Poisson ratio $$\sigma$$, bulk modulus $$(K)$$ and modulus of rigidity $$\eta$$ of a given solid object:

We need to identify the correct relationship between Poisson ratio $$\sigma$$, bulk modulus $$K$$, and modulus of rigidity $$\eta$$.

Standard relation:

The relationship between Young's modulus $$Y$$, bulk modulus $$K$$, and Poisson's ratio $$\sigma$$ is:

$$Y = 3K(1 - 2\sigma)$$ ... (i)

The relationship between Young's modulus $$Y$$, modulus of rigidity $$\eta$$, and Poisson's ratio $$\sigma$$ is:

$$Y = 2\eta(1 + \sigma)$$ ... (ii)

Eliminating Y:

From (i) and (ii):

$$3K(1 - 2\sigma) = 2\eta(1 + \sigma)$$

$$3K - 6K\sigma = 2\eta + 2\eta\sigma$$

$$3K - 2\eta = 6K\sigma + 2\eta\sigma$$

$$3K - 2\eta = \sigma(6K + 2\eta)$$

$$\sigma = \frac{3K - 2\eta}{6K + 2\eta}$$

The correct answer is Option 1: $$\sigma = \frac{3K - 2\eta}{6K + 2\eta}$$.