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

Options

A. `[p^(1)A^(-1)t^(-1)]`
B. `[p^(2)A^(1)t^(1)]`
C. `[p^(1)A^(1//2)t^(1)]`
D. `[p^(1)A^(1//2)t^(-1)]`

Explanation

To find the dimensional formula of energy (E) when momentum (p), area (A), and time (t) are taken as fundamental quantities, we can follow these steps:

### Step 1: Write the relationship for energy
We start by expressing energy in terms of the fundamental quantities:
\[ E = k \cdot p^a \cdot A^b \cdot t^c \]
where \( k \) is a constant and \( a, b, c \) are the powers we need to determine.

### Step 2: Write the dimensional formulas for the quantities
The dimensional formulas for the quantities are:
- Momentum \( p = [M][L][T^{-1}] \)
- Area \( A = [L^2] \)
- Time \( t = [T] \)

Thus, we can write:
- \( [p] = [M][L][T^{-1}] \)
- \( [A] = [L^2] \)
- \( [t] = [T] \)

### Step 3: Substitute the dimensional formulas into the equation
Now, substituting the dimensional formulas into our equation for energy:
\[ [E] = [M^0][L^0][T^0] = [M^1][L^2][T^{-2}] \]
This is the dimensional formula for energy.

### Step 4: Express the right-hand side in terms of dimensions
Now, substituting the dimensions of \( p, A, \) and \( t \) into the equation:
\[ [E] = [p^a][A^b][t^c] \]
\[ = ([M][L][T^{-1}])^a \cdot ([L^2])^b \cdot ([T])^c \]
\[ = [M^a][L^{a + 2b}][T^{-a + c}] \]

### Step 5: Equate the dimensions
Now, we equate the dimensions from both sides:
1. For mass: \( a = 1 \)
2. For length: \( a + 2b = 2 \)
3. For time: \( -a + c = -2 \)

### Step 6: Solve the equations
From \( a = 1 \):
- Substitute \( a = 1 \) into \( a + 2b = 2 \):
\[ 1 + 2b = 2 \]
\[ 2b = 1 \]
\[ b = \frac{1}{2} \]

- Substitute \( a = 1 \) into \( -a + c = -2 \):
\[ -1 + c = -2 \]
\[ c = -1 \]

### Step 7: Write the final dimensional formula
Now we can write the dimensional formula for energy:
\[ [E] = p^1 \cdot A^{1/2} \cdot t^{-1} \]
Thus, the dimensional formula of energy is:
\[ [E] = [p^1][A^{1/2}][t^{-1}] \]

### Final Answer
The dimensional formula of energy when momentum, area, and time are taken as fundamental quantities is:
\[ [E] = [p^1][A^{1/2}][t^{-1}] \]