If energy (E), velocity (v) and time (T) are fundamental units. What will be the dimension of surface tension?

Explanation

To find the dimension of surface tension when energy (E), velocity (v), and time (T) are considered as fundamental units, we can follow these steps:

### Step 1: Understand Surface Tension
Surface tension (S) is defined as the force per unit length. Therefore, its dimensional formula can be expressed as:
\[ S = \frac{F}{L} \]
where \(F\) is force and \(L\) is length.

### Step 2: Determine the Dimension of Force
Force (F) can be expressed using Newton's second law:
\[ F = m \cdot a \]
where \(m\) is mass and \(a\) is acceleration. The dimension of mass is \(M\) and the dimension of acceleration is:
\[ a = \frac{L}{T^2} \]
Thus, the dimension of force is:
\[ F = M \cdot \frac{L}{T^2} = M L T^{-2} \]

### Step 3: Determine the Dimension of Length
The dimension of length is simply:
\[ L = L \]

### Step 4: Combine to Find the Dimension of Surface Tension
Now substituting the dimensions of force and length into the expression for surface tension:
\[ S = \frac{F}{L} = \frac{M L T^{-2}}{L} = M L^{0} T^{-2} \]
Thus, the dimension of surface tension is:
\[ S = M L^{0} T^{-2} \]

### Step 5: Express Surface Tension in Terms of E, V, and T
We can express surface tension as proportional to energy (E), velocity (V), and time (T):
\[ S \propto E^X V^Y T^Z \]
We need to find the values of \(X\), \(Y\), and \(Z\).

### Step 6: Determine the Dimensions of Energy, Velocity, and Time
- The dimension of energy (E) is given by:
\[ E = F \cdot d = (M L T^{-2}) \cdot L = M L^{2} T^{-2} \]
- The dimension of velocity (V) is:
\[ V = \frac{L}{T} = L T^{-1} \]
- The dimension of time (T) is:
\[ T = T \]

### Step 7: Substitute Dimensions into the Equation
Substituting the dimensions into the equation:
\[ M^{1} L^{0} T^{-2} = (M L^{2} T^{-2})^X \cdot (L T^{-1})^Y \cdot (T)^Z \]
This expands to:
\[ M^{1} L^{0} T^{-2} = M^{X} L^{2X + Y} T^{-2X - Y + Z} \]

### Step 8: Equate the Powers of Each Dimension
Now we equate the powers of \(M\), \(L\), and \(T\):
1. For \(M\): \(X = 1\)
2. For \(L\): \(2X + Y = 0\)
3. For \(T\): \(-2X - Y + Z = -2\)

### Step 9: Solve the Equations
From \(X = 1\):
- Substitute \(X\) into \(2X + Y = 0\):
\[ 2(1) + Y = 0 \implies Y = -2 \]
- Substitute \(X\) and \(Y\) into \(-2X - Y + Z = -2\):
\[ -2(1) - (-2) + Z = -2 \implies -2 + 2 + Z = -2 \implies Z = -2 \]

### Step 10: Write the Final Expression for Surface Tension
Now substituting the values of \(X\), \(Y\), and \(Z\) back into the proportionality:
\[ S \propto E^{1} V^{-2} T^{-2} \]
Thus, we can express surface tension as:
\[ S = K \cdot \frac{E}{V^2 T^2} \]
where \(K\) is a constant.

### Final Answer
The dimension of surface tension in terms of energy, velocity, and time is:
\[ S \propto \frac{E}{V^2 T^2} \]