If energy `E`, velocity `v` and time `T` are taken as fundamental quanties, the dimensional formula for surface tension is

Options

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

Answer

To find the dimensional formula for surface tension when energy \(E\), velocity \(v\), and time \(T\) are taken as fundamental quantities, we can follow these steps:

### Step 1: Understand the definition of surface tension
Surface tension (\(S\)) is defined as the force per unit length. The dimensional formula for force is given by:
\[ \text{Force} = \text{mass} \times \text{acceleration} = [M][L][T^{-2}] = MLT^{-2} \]
Thus, the dimensional formula for surface tension can be expressed as:
\[ S = \frac{\text{Force}}{\text{Length}} = \frac{MLT^{-2}}{L} = MT^{-2} \]

### Step 2: Express surface tension in terms of fundamental quantities
Since we are given that energy \(E\), velocity \(v\), and time \(T\) are the fundamental quantities, we need to express surface tension in terms of these quantities.

### Step 3: Write the dimensional formula of energy and velocity
The dimensional formula for energy \(E\) is:
\[ E = [M][L^2][T^{-2}] = ML^2T^{-2} \]
The dimensional formula for velocity \(v\) is:
\[ v = [L][T^{-1}] = LT^{-1} \]

### Step 4: Set up the relationship
We can express surface tension \(S\) as:
\[ S = k E^a v^b T^c \]
where \(k\) is a constant and \(a\), \(b\), and \(c\) are the powers we need to determine.

### Step 5: Substitute the dimensions
Substituting the dimensions into the equation gives:
\[ MT^{-2} = k (ML^2T^{-2})^a (LT^{-1})^b (T)^c \]
This simplifies to:
\[ MT^{-2} = k M^a L^{2a+b} T^{-2a-b+c} \]

### Step 6: Compare dimensions
Now we can compare the dimensions on both sides:
1. For mass: \(1 = a\)
2. For length: \(0 = 2a + b\)
3. For time: \(-2 = -2a - b + c\)

### Step 7: Solve the equations
From the first equation, we have:
\[ a = 1 \]
Substituting \(a = 1\) into the second equation:
\[ 0 = 2(1) + b \implies b = -2 \]
Now substituting \(a = 1\) and \(b = -2\) into the third equation:
\[ -2 = -2(1) - (-2) + c \implies -2 = -2 + 2 + c \implies c = -2 \]

### Step 8: Write the final expression
Now substituting \(a\), \(b\), and \(c\) back into the expression for surface tension:
\[ S = k E^1 v^{-2} T^{-2} \]
Since \(k = 1\), we have:
\[ S = E v^{-2} T^{-2} \]

### Step 9: Final dimensional formula
Thus, the dimensional formula for surface tension is:
\[ S = E^1 v^{-2} T^{-2} \implies S = [M^1 L^2 T^{-2}]^1 [L T^{-1}]^{-2} [T]^{-2} = M^1 L^{2-2} T^{-2-2} = M^1 L^0 T^{-4} = MT^{-4} \]

### Final Answer
The dimensional formula for surface tension is:
\[ [M T^{-2}] \]