Choose the correct relationship between Poisson ratio `(sigma)`,Bulk modulus (K) and modulus of rigidity `(eta)` of given solid object:

Options

A. `sigma=(3K+2eta)/(6K+2eta)`
B. `sigma=(6K-2eta)/(3K-2eta)`
C. `sigma=(3K-2eta)/(6K+2eta)`
D. `sigma=(6K-2eta)/(3K-2eta)`

Answer

To find the correct relationship between the Poisson ratio (σ), Bulk modulus (K), and modulus of rigidity (η) of a solid object, we can use the known relationships between these quantities.

### Step 1: Write down the relevant equations
We know two important relationships:
1. \( \gamma = 2\eta(1 + \sigma) \) (where γ is the modulus of elasticity)
2. \( \gamma = 3K(1 - 2\sigma) \)

### Step 2: Set the two expressions for γ equal to each other
Since both expressions equal γ, we can equate them:
\[ 2\eta(1 + \sigma) = 3K(1 - 2\sigma) \]

### Step 3: Expand both sides
Expanding both sides gives:
\[ 2\eta + 2\eta\sigma = 3K - 6K\sigma \]

### Step 4: Rearrange the equation
Rearranging the equation to group terms involving σ:
\[ 2\eta + 2\eta\sigma + 6K\sigma = 3K \]
\[ 2\eta\sigma + 6K\sigma = 3K - 2\eta \]

### Step 5: Factor out σ
Factoring out σ from the left side:
\[ \sigma(2\eta + 6K) = 3K - 2\eta \]

### Step 6: Solve for σ
Now, we can solve for σ:
\[ \sigma = \frac{3K - 2\eta}{2\eta + 6K} \]

### Step 7: Identify the correct option
Now, we can compare this derived expression with the given options in the question. The expression we derived matches with option 3.

### Final Answer
Thus, the correct relationship is:
\[ \sigma = \frac{3K - 2\eta}{2\eta + 6K} \]