The dimensional formula of pressure gradient is

Options

A. `[MLT^(-2)]`
B. `[ML^(2)T^(-2)]`
C. `[ML^(-2)T^(-2)]`
D. `[MLT^(-1)]`

Answer

To find the dimensional formula of pressure gradient, we can follow these steps:

### Step 1: Understand the concept of pressure gradient
The pressure gradient is defined as the change in pressure per unit length. It can be mathematically expressed as:
\[ \text{Pressure Gradient} (Pg) = \frac{\Delta P}{\Delta x} \]
where \( \Delta P \) is the change in pressure and \( \Delta x \) is the change in length.

### Step 2: Determine the dimensional formula of pressure
Pressure is defined as force per unit area. The formula for pressure \( P \) is:
\[ P = \frac{F}{A} \]
where \( F \) is force and \( A \) is area.

The dimensional formula for force \( F \) is:
\[ [F] = [M][L][T^{-2}] \]
where \( M \) is mass, \( L \) is length, and \( T \) is time.

The area \( A \) can be expressed in terms of length:
\[ [A] = [L^2] \]

Thus, the dimensional formula for pressure becomes:
\[ [P] = \frac{[M][L][T^{-2}]}{[L^2]} = [M][L^{-1}][T^{-2}] \]

### Step 3: Determine the dimensional formula of pressure gradient
Now, substituting the dimensional formula of pressure into the formula for pressure gradient:
\[ Pg = \frac{\Delta P}{\Delta x} \]

The dimensional formula for \( \Delta x \) (length) is:
\[ [\Delta x] = [L] \]

Thus, the dimensional formula for pressure gradient is:
\[ [Pg] = \frac{[M][L^{-1}][T^{-2}]}{[L]} = [M][L^{-2}][T^{-2}] \]

### Final Result
The dimensional formula of pressure gradient is:
\[ [M][L^{-2}][T^{-2}] \]