A) velocity gradient
B) potential gradient
C) energy gradient
D) none of these
Correct Answer: C
Solution :
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 \[\left( \frac{\Delta P}{\Delta x} \right)=\frac{[ML{{T}^{2}}]/[{{L}^{2}}]}{[L]}\] \[=[M{{L}^{-2}}{{T}^{-2}}]\] Dimension of velocity gradient \[=\left[ -\frac{\Delta v}{\Delta x} \right]=\frac{m/s}{m}=\frac{[L{{T}^{-1}}]}{[L]}\] \[=[{{M}^{0}}{{L}^{0}}{{T}^{-1}}]\] Dimensions of potential gradient \[=\left( -\frac{\Delta V}{\Delta x} \right)\frac{\Delta W/Q}{\Delta x}=\frac{[ML{{T}^{-2}}][L]}{[AT][L]}\] \[=[ML{{T}^{-3}}{{A}^{-1}}]\] Energy gradient\[=-\frac{\Delta E}{\Delta x}=\frac{Nm}{m}\] \[\therefore \] Dimensions of \[\left( \frac{\Delta E}{\Delta x} \right)=\frac{[ML{{T}^{-2}}][L]}{[L]}\] \[=[ML{{T}^{-2}}]\] As observed from above results we see that none of the dimensions are same as of pressure gradient hence, option (c) is correct.
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