If energy `(E )` , velocity `(V)` and time `(T)` are chosen as the fundamental quantities , the dimensions formula of surface tension will be

Options

A. `[EV^(-2)T^(-1)]`
B. `[EV^(-1)T^(-2)]`
C. `[EV^(-2)T^(-2)]`
D. `[E^(-2)V^(-1)T^(-3)]`

Explanation

To find the dimensional formula of surface tension using energy (E), velocity (V), and time (T) as fundamental quantities, we can follow these steps:

### Step 1: Understand the definition of surface tension
Surface tension (σ) is defined as the force per unit length. Its dimensional formula can be expressed as:
\[ \sigma = \frac{F}{L} \]
where \( F \) is force and \( L \) is length.

### Step 2: Write the dimensions of force and length
The dimension of force (F) can be derived from Newton's second law:
\[ F = m \cdot a \]
where \( a \) (acceleration) has dimensions of \( \frac{L}{T^2} \). Therefore, the dimension of force is:
\[ [F] = [M][L][T^{-2}] = M^1 L^1 T^{-2} \]

The dimension of length (L) is simply:
\[ [L] = L^1 \]

### Step 3: Substitute the dimensions into the surface tension formula
Now substituting the dimensions of force and length into the surface tension formula:
\[ [\sigma] = \frac{[F]}{[L]} = \frac{M^1 L^1 T^{-2}}{L^1} = M^1 L^0 T^{-2} \]

### Step 4: Express surface tension in terms of E, V, and T
We assume that surface tension can be expressed in terms of energy (E), velocity (V), and time (T) as:
\[ \sigma \propto E^x V^y T^z \]

### Step 5: Write the dimensions of energy, velocity, and time
- The dimension of energy (E) is:
\[ [E] = M^1 L^2 T^{-2} \]
- The dimension of velocity (V) is:
\[ [V] = M^0 L^1 T^{-1} \]
- The dimension of time (T) is:
\[ [T] = M^0 L^0 T^1 \]

### Step 6: Substitute the dimensions into the equation
Substituting the dimensions into the equation:
\[ M^1 L^0 T^{-2} = (M^1 L^2 T^{-2})^x (M^0 L^1 T^{-1})^y (M^0 L^0 T^1)^z \]

### Step 7: Expand and equate the powers
Expanding the right-hand side:
\[ M^{x} L^{2x+y} T^{-2x-y+z} \]
Now equate the powers of M, L, and T from both sides:
1. For M: \( x = 1 \)
2. For L: \( 2x + y = 0 \)
3. For T: \( -2x - y + z = -2 \)

### Step 8: Solve the equations
From the first equation, we have:
\[ x = 1 \]
Substituting \( x = 1 \) into the second equation:
\[ 2(1) + y = 0 \implies y = -2 \]
Substituting \( x = 1 \) and \( y = -2 \) into the third equation:
\[ -2(1) - (-2) + z = -2 \implies -2 + 2 + z = -2 \implies z = -2 \]

### Step 9: Write the final expression for surface tension
Thus, we have:
\[ x = 1, \quad y = -2, \quad z = -2 \]
So, the dimensional formula for surface tension is:
\[ \sigma \propto E^1 V^{-2} T^{-2} \]

### Final Answer
The dimensions of surface tension in terms of energy, velocity, and time are:
\[ \sigma \propto E^1 V^{-2} T^{-2} \]