If area(A), time(T) and momentum(P) is assume as fundamental quantities then dimensional formula of energy will be

Options

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

Answer

To solve the problem of finding the dimensional formula of energy in terms of area (A), time (T), and momentum (P), we will follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We start with the assumption that energy (E) can be expressed in terms of the fundamental quantities area (A), time (T), and momentum (P). We can write:
\[ E \propto A^x T^y P^z \]
where \(x\), \(y\), and \(z\) are the powers we need to determine.

2. **Writing the Dimensional Formulas**:
- The dimensional formula for area \(A\) is \(L^2\).
- The dimensional formula for time \(T\) is \(T^1\).
- The dimensional formula for momentum \(P\) is \(MLT^{-1}\).

3. **Expressing Energy's Dimensional Formula**:
The dimensional formula of energy in terms of mass (M), length (L), and time (T) is:
\[ [E] = ML^2T^{-2} \]

4. **Substituting the Dimensional Formulas**:
Substituting the dimensional formulas into our proportionality:
\[ [E] = A^x T^y P^z = (L^2)^x (T^1)^y (MLT^{-1})^z \]
This expands to:
\[ [E] = L^{2x} T^y M^z L^z T^{-z} = M^z L^{2x + z} T^{y - z} \]

5. **Comparing Dimensions**:
Now we compare the dimensions of both sides:
\[ ML^2T^{-2} = M^z L^{2x + z} T^{y - z} \]
This gives us three equations:
- For mass (M): \(z = 1\)
- For length (L): \(2x + z = 2\)
- For time (T): \(y - z = -2\)

6. **Solving the Equations**:
From \(z = 1\):
- Substitute \(z = 1\) into \(2x + z = 2\):
\[ 2x + 1 = 2 \implies 2x = 1 \implies x = \frac{1}{2} \]
- Substitute \(z = 1\) into \(y - z = -2\):
\[ y - 1 = -2 \implies y = -1 \]

7. **Final Values**:
We have determined:
- \(x = \frac{1}{2}\)
- \(y = -1\)
- \(z = 1\)

8. **Writing the Dimensional Formula of Energy**:
Therefore, the dimensional formula of energy in terms of area, time, and momentum is:
\[ E = A^{\frac{1}{2}} T^{-1} P^1 \]

### Conclusion:
The dimensional formula of energy in terms of area (A), time (T), and momentum (P) is:
\[ E = A^{\frac{1}{2}} T^{-1} P^1 \]