Which of the following equations is dimensionally correct?

Options

A. pressure = Engery per units volume
B. pressure = Energy per units area
C. pressure = Momentum `xx` Volume `xx` time
D. pressure = Force `xx` area

Answer

To determine which of the given equations is dimensionally correct, we need to analyze each option based on the dimensions of the physical quantities involved. Let's go through the steps systematically.

### Step-by-Step Solution:

1. **Understand the Dimensions of Pressure:**
- Pressure (P) is defined as force (F) per unit area (A).
- The dimension of force (F) is given by \( [F] = MLT^{-2} \).
- The dimension of area (A) is given by \( [A] = L^2 \).
- Therefore, the dimension of pressure is:
\[ [P] = \frac{[F]}{[A]} = \frac{MLT^{-2}}{L^2} = ML^{-1}T^{-2} \]

2. **Examine Each Option:**

**Option A: \( E/V \)**
- The dimension of energy (E) is \( [E] = M L^2 T^{-2} \).
- The dimension of volume (V) is \( [V] = L^3 \).
- Now, calculate the dimension of \( E/V \):
\[ \frac{[E]}{[V]} = \frac{ML^2T^{-2}}{L^3} = ML^{-1}T^{-2} \]
- This matches the dimension of pressure. Thus, **Option A is dimensionally correct**.

**Option B: \( E/A \)**
- Using the same dimension for energy (E) and area (A):
\[ \frac{[E]}{[A]} = \frac{ML^2T^{-2}}{L^2} = MT^{-2} \]
- This does not match the dimension of pressure. Thus, **Option B is incorrect**.

**Option C: \( P = p \cdot V \cdot T \)**
- The dimension of momentum (p) is \( [p] = MLT^{-1} \).
- The dimension of volume (V) is \( [V] = L^3 \).
- The dimension of time (T) is \( [T] = T \).
- Now, calculate the dimension of \( p \cdot V \cdot T \):
\[ [p \cdot V \cdot T] = [p] \cdot [V] \cdot [T] = (MLT^{-1}) \cdot (L^3) \cdot (T) = ML^4T^{0} \]
- This does not match the dimension of pressure. Thus, **Option C is incorrect**.

**Option D: \( P = F \cdot A \)**
- This is incorrect by definition, as pressure is defined as force per unit area, not force multiplied by area. Thus, **Option D is incorrect**.

3. **Conclusion:**
- The only dimensionally correct equation is **Option A: \( E/V \)**.