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

### Step 1: Write the dimensional formulas for the fundamental quantities
1. **Area (A)**: The area is given by length squared, so its dimensional formula is:
\[ [A] = L^2 \]

2. **Time (T)**: The dimensional formula for time is:
\[ [T] = T^1 \]

3. **Momentum (P)**: Momentum is defined as mass times velocity. The dimensional formula for mass is \(M\) and for velocity (length/time) is \(L T^{-1}\). Therefore, the dimensional formula for momentum is:
\[ [P] = M L T^{-1} \]

### Step 2: Assume the dimensional formula of energy in terms of A, T, and P
Assume the dimensional formula of energy can be expressed as:
\[ [E] = A^a T^b P^c \]
where \(a\), \(b\), and \(c\) are the powers to be determined.

### Step 3: Substitute the dimensional formulas into the equation
Substituting the dimensional formulas we have:
\[ [E] = (L^2)^a (T^1)^b (M L T^{-1})^c \]
This expands to:
\[ [E] = L^{2a} T^b M^c L^c T^{-c} \]
Combining the terms gives:
\[ [E] = M^c L^{2a+c} T^{b-c} \]

### Step 4: Write the known dimensional formula of energy
The known dimensional formula for energy is:
\[ [E] = M^1 L^2 T^{-2} \]

### Step 5: Set up equations by comparing powers
From the equations:
1. For mass \(M\):
\[ c = 1 \]

2. For length \(L\):
\[ 2a + c = 2 \]

3. For time \(T\):
\[ b - c = -2 \]

### Step 6: Solve the equations
1. From \(c = 1\), substitute into the second equation:
\[ 2a + 1 = 2 \implies 2a = 1 \implies a = \frac{1}{2} \]

2. Substitute \(c = 1\) into the third equation:
\[ b - 1 = -2 \implies b = -1 \]

### Step 7: Write the final dimensional formula of energy
Now substituting the values of \(a\), \(b\), and \(c\) back into the expression for energy:
\[ [E] = A^{\frac{1}{2}} T^{-1} P^1 \]

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