Question

Which of the following equations is dimensionally incorrect?
Where $t=$ time, $h=$ height, $s=$ surface tension, $\theta =$ angle,$\rho =$ density $,a,r=$ radius, $g=$ acceleration due to gravity, $v=$ volume, $p=$ pressure, $W=$ work done, $\Gamma =$ torque, $\in =$ permittivity, $E=$ electric field, $J=$ current density, $L=$ length

• A. $v=\frac{\pi p{a}^4}{8\eta L}$
• B. $h=\frac{2\text{scos}\theta }{\rho rg}$
• C. $J=\in \frac{\partial E}{\partial t}$
• D. $W=\Gamma \theta$

Answer 1

Answer: (A)

(i) $\frac{\pi p{a}^4}{8\eta L}=\frac{dv}{dt}$
$=$ Volumetric flow rate
(poiseuille’s law)
(ii) $h\rho g=\frac{2s}{r}\cos \theta$
(iii) RHS $\Rightarrow \epsilon \times \frac{1}{4\pi {\epsilon }_0}\frac{a}{r^2}\times \frac{1}{\epsilon }=\frac{q}{t}\times \frac{1}{r^2}$
$=\frac{I}{L^2}=I{L}^{-2}$
LHS
$T=\frac{I}{A}=I{L}^{-2}$
(iv) $W=\tau \theta$