In the formula `V = E^b d^a`, if V , E and d are the velocity of longitudinal waves, bulk modulus of elasticity and density of the gaseous medium respectively, then the values of a and b are respectively.

Options

A. `- 1/2 and 1/2`
B. `1/2 and -1/2`
C. `- 1/(sqrt(2)) and 1/(sqrt2)`
D. `- 1/(sqrt2) and - 1/(sqrt2)`

Explanation

To solve the problem given by the formula \( V = E^b d^a \), where \( V \) is the velocity of longitudinal waves, \( E \) is the bulk modulus of elasticity, and \( d \) is the density of the gaseous medium, we need to find the values of \( a \) and \( b \).

### Step 1: Write the dimensions of each variable
- **Velocity (V)**: The dimension of velocity is given by:
\[ [V] = L T^{-1} \]
- **Bulk Modulus of Elasticity (E)**: The bulk modulus is defined as stress divided by strain. Since strain is dimensionless, we have:
\[ [E] = \frac{\text{Force}}{\text{Area}} = \frac{M L T^{-2}}{L^2} = M L^{-1} T^{-2} \]
- **Density (d)**: The density is mass per unit volume:
\[ [d] = \frac{M}{L^3} = M L^{-3} \]

### Step 2: Substitute the dimensions into the equation
The equation \( V = E^b d^a \) can be expressed in terms of dimensions:
\[ [L T^{-1}] = [E]^b [d]^a \]
Substituting the dimensions we found:
\[ L T^{-1} = (M L^{-1} T^{-2})^b (M L^{-3})^a \]

### Step 3: Expand the right-hand side
Expanding the right-hand side:
\[ L T^{-1} = M^b L^{-b} T^{-2b} \cdot M^a L^{-3a} = M^{b+a} L^{-b-3a} T^{-2b} \]

### Step 4: Equate the dimensions
Now, we equate the dimensions on both sides:
1. For mass (M):
\[ 0 = b + a \quad \text{(1)} \]
2. For length (L):
\[ 1 = -b - 3a \quad \text{(2)} \]
3. For time (T):
\[ -1 = -2b \quad \text{(3)} \]

### Step 5: Solve the equations
From equation (3):
\[ -1 = -2b \implies b = \frac{1}{2} \]

Substituting \( b = \frac{1}{2} \) into equation (1):
\[ 0 = \frac{1}{2} + a \implies a = -\frac{1}{2} \]

### Conclusion
Thus, the values of \( a \) and \( b \) are:
\[ a = -\frac{1}{2}, \quad b = \frac{1}{2} \]

### Final Answer
The values of \( a \) and \( b \) are respectively \( -\frac{1}{2} \) and \( \frac{1}{2} \).