The potential energy of a particle varies with distance `x` from a fixed origin as `U = (A sqrt(x))/( x^(2) + B)`, where `A and B` are dimensional constants , then find the dimensional formula for `AB`.

Options

A. `[ML^(11//2)T^(-1)]`
B. `[ML^(11)T^(-2)]`
C. `[ML^(7//2)T^(-2)]`
D. `[ML^(11//2)T^(-2)]`

Explanation

To find the dimensional formula for the product \( AB \) given the potential energy \( U = \frac{A \sqrt{x}}{x^2 + B} \), we will follow these steps:

### Step 1: Identify the dimensions of potential energy \( U \)
The dimensional formula for energy is given by:
\[ [U] = [\text{Energy}] = [M][L^2][T^{-2}] \]
Thus,
\[ [U] = M L^2 T^{-2} \]

### Step 2: Analyze the expression for potential energy
The expression for potential energy is:
\[ U = \frac{A \sqrt{x}}{x^2 + B} \]
In this expression, \( x \) represents distance, which has the dimension:
\[ [x] = [L] \]
Therefore, we can express \( \sqrt{x} \) as:
\[ [\sqrt{x}] = [L^{1/2}] \]

### Step 3: Determine the dimensions of \( B \)
The term \( x^2 + B \) must have the same dimensions as \( x^2 \) because they are being added. Thus:
\[ [x^2] = [L^2] \]
This implies that:
\[ [B] = [L^2] \]

### Step 4: Substitute dimensions into the expression for \( U \)
Now, substituting the dimensions into the expression for \( U \):
\[ U = \frac{A [L^{1/2}]}{[L^2] + [L^2]} \]
The denominator simplifies to \( [L^2] \), so we have:
\[ U = \frac{A [L^{1/2}]}{[L^2]} = A \frac{[L^{1/2}]}{[L^2]} = A [L^{-3/2}] \]

### Step 5: Equate dimensions of \( U \) and solve for \( A \)
Since we know that \( [U] = M L^2 T^{-2} \), we can equate:
\[ A [L^{-3/2}] = M L^2 T^{-2} \]
From this, we can find the dimensions of \( A \):
\[ [A] = M L^2 T^{-2} [L^{3/2}] = M L^{2 + 3/2} T^{-2} = M L^{7/2} T^{-2} \]

### Step 6: Find the product \( AB \)
Now we have:
\[ [A] = M L^{7/2} T^{-2} \quad \text{and} \quad [B] = L^2 \]
To find \( AB \):
\[ [AB] = [A][B] = (M L^{7/2} T^{-2})(L^2) = M L^{7/2 + 2} T^{-2} = M L^{11/2} T^{-2} \]

### Final Answer
The dimensional formula for \( AB \) is:
\[ [AB] = M L^{11/2} T^{-2} \]