To find the dimensional formula of surface tension when energy (E), velocity (V), and time (T) are chosen 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 derived from Newton's second law, which states that force (F) is mass (m) times acceleration (a).
### Step 2: Write the dimensional formula for force
The dimensional formula for force (F) is:
\[
[F] = [M][L][T^{-2}]
\]
where:
- [M] is the dimension of mass,
- [L] is the dimension of length,
- [T] is the dimension of time.
### Step 3: Write the dimensional formula for surface tension
Since surface tension is force per unit length, we can express it as:
\[
[S] = \frac{[F]}{[L]} = \frac{[M][L][T^{-2}]}{[L]} = [M][T^{-2}]
\]
### Step 4: Express surface tension in terms of E, V, and T
Now, we need to express the dimensions of surface tension using the fundamental quantities E, V, and T. We assume:
\[
[S] \propto E^a V^b T^c
\]
where \( a \), \( b \), and \( c \) are the powers we need to determine.
### Step 5: Write the dimensional formulas for E, V, and T
The dimensional formulas for the chosen quantities are:
- Energy (E):
\[
[E] = [M][L^2][T^{-2}]
\]
- Velocity (V):
\[
[V] = [L][T^{-1}]
\]
- Time (T):
\[
[T] = [T]
\]
### Step 6: Set up the equation for dimensions
Substituting these into our expression for surface tension gives:
\[
[M][T^{-2}] = [M^{a}][L^{2a}][T^{-2a}][L^{b}][T^{-b}][T^{c}]
\]
### Step 7: Equate dimensions for M, L, and T
Now we can equate the coefficients for each dimension:
1. For mass (M):
\[
1 = a
\]
2. For length (L):
\[
0 = 2a + b
\]
3. For time (T):
\[
-2 = -2a - b + c
\]
### Step 8: Solve the equations
From \( 1 = a \), we have:
\[
a = 1
\]
Substituting \( a = 1 \) into the second equation:
\[
0 = 2(1) + b \implies b = -2
\]
Substituting \( a = 1 \) and \( b = -2 \) into the third equation:
\[
-2 = -2(1) - (-2) + c \implies -2 = -2 + 2 + c \implies c = -2
\]
### Step 9: Write the final dimensional formula
Thus, we can express surface tension in terms of E, V, and T as:
\[
[S] = E^1 V^{-2} T^{-2}
\]
### Final Answer:
The dimensional formula for surface tension is:
\[
S = E^1 V^{-2} T^{-2}
\]
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