Which of the following equations is dimensionally incorrect?
Where $t$ = time, $h$ = height, $s$ = surface tension, $\theta$ = angle, $\rho$ = density, $a, r$ = radius, $g$ = the acceleration due to gravity, $V$ = volume, $p$ = pressure, $W$ = work done, $\tau$ = torque, $\epsilon$ = permittivity, $E$ = electric field, $J$ = current density, $L$ = length.
Answer: (B) $V = \dfrac{\pi pa^4}{8\eta L}$
(1) Torque × angle has the dimensions of work. Correct.
(2) $\dfrac{\pi pa^4}{8\eta L}$ is Poiseuille's expression for the volume flow rate (volume per second, $L^3T^{-1}$), not for volume $V$ ($L^3$). Incorrect.
(3) Capillary rise formula. Correct.
(4) $\epsilon\dfrac{\partial E}{\partial t}$ is the displacement current density. Correct.
Solution by Sreeraj P, M.Sc Physics