If momentum `(p)`, area `(A)` and time`(t) `are taken to be fundamental quantities then energy has the dimensional formula

Options

A. `[pA^(-1)T^(-1)]`
B. `[P^(2)AT]`
C. `[pA^(-1//2)T]`
D. `[pA^(1//2)T^(-1)]`

Explanation

To find the dimensional formula of energy (E) in terms of momentum (p), area (A), and time (t), we can follow these steps:

### Step 1: Understand the definitions of the quantities
- **Momentum (p)** is defined as the product of mass (m) and velocity (v). Therefore, the dimensional formula of momentum is:
\[ [p] = [m][v] = [m][L][T^{-1}] = mL T^{-1} \]

- **Area (A)** is defined as length squared. Therefore, the dimensional formula of area is:
\[ [A] = [L^2] \]

- **Time (t)** has the dimensional formula:
\[ [t] = [T] \]

### Step 2: Write the relationship for energy
Energy can be expressed in terms of momentum, area, and time as:
\[ E \propto p^A A^B t^C \]
where A, B, and C are the powers to which momentum, area, and time are raised, respectively.

### Step 3: Substitute the dimensional formulas
Substituting the dimensional formulas into the equation gives:
\[ [E] \propto (mL T^{-1})^A (L^2)^B (T)^C \]

### Step 4: Expand the equation
Expanding this, we get:
\[ [E] \propto m^A L^{A + 2B} T^{-A + C} \]

### Step 5: Set the dimensional formula of energy
The dimensional formula of energy (E) is known to be:
\[ [E] = mL^2 T^{-2} \]

### Step 6: Equate the powers of dimensions
Now, we can equate the powers of each dimension from both sides:
1. For mass (m):
\[ A = 1 \]

2. For length (L):
\[ A + 2B = 2 \]

3. For time (T):
\[ -A + C = -2 \]

### Step 7: Solve the equations
From the first equation, we have:
\[ A = 1 \]

Substituting \(A = 1\) into the second equation:
\[ 1 + 2B = 2 \implies 2B = 1 \implies B = \frac{1}{2} \]

Substituting \(A = 1\) into the third equation:
\[ -1 + C = -2 \implies C = -1 \]

### Step 8: Write the final expression for energy
Now substituting the values of A, B, and C back into the expression for energy:
\[ E \propto p^1 A^{\frac{1}{2}} t^{-1} \]

Thus, the dimensional formula of energy in terms of momentum, area, and time is:
\[ E \propto p^1 A^{\frac{1}{2}} t^{-1} \]