Question

Pressure gradient has the same dimensions as that of:

• A. velocity gradient
• B. potential gradient
• C. energy gradient
• D. none of these

Answer 1

Answer: (C)

In order to arrive at the correct answer we solve for dimensional formula of each individually.
Pressure gradient $=-\frac{\Delta P}{\Delta x}=\frac{N/m^2}{m}$
$\therefore$ Dimensions of $\left(\frac{\Delta P}{\Delta x}\right)=\frac{\left[ML{T}^{-2}\right]/\left[L^2\right]}{[L]}$
$=\left[M{L}^{-2}{T}^{-2}\right]$
Dimension of velocity gradient
$=\left[-\frac{\Delta v}{\Delta x}\right]=\frac{m/s}{m}=\frac{\left[L{T}^{-1}\right]}{[L]}$
$=\left[M^0{L}^0{T}^{-1}\right]$
Dimensions of potential gradient
$=\left(-\frac{\Delta V}{\Delta x}\right)\frac{\Delta W/Q}{\Delta x}=\frac{\left[ML{T}^{-2}\right][L]}{[AT][L]}$
$=\left[ML{T}^{-3}{A}^{-1}\right]$
Energy gradient $=-\frac{\Delta E}{\Delta x}=\frac{Nm}{m}$
Dimensions of $\left(\frac{\Delta E}{\Delta x}\right)=\frac{\left[ML{T}^{-2}\right][L]}{[L]}$
$=\left[ML{T}^{-2}\right]$
As observed from above results we see that none of the dimensions are same as of pressure gradient