Which of the following equations is dimensionally incorrect? Wher… - Vidyadip

MCQ

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, $\varepsilon=$ permittivity, ${E}=$ electric field, ${J}=$ current density, ${L}=$ length.

✓
${v}=\frac{\pi {pa}^{4}}{8 \eta {L}}$
B
${h}=\frac{2 {s} \cos \theta}{\rho {rg}}$
C
${J}=\varepsilon \frac{\partial {E}}{\partial {t}}$
D
${W}=\Gamma \theta$

✓

Answer

Correct option: A.

${v}=\frac{\pi {pa}^{4}}{8 \eta {L}}$

a
$(i)$ $\frac{\pi {pa}^{4}}{8 \eta {L}}=\frac{{d} {v}}{{dt}}=$ Volumetric flow rate

(poiseuille's law)

$(ii)$ ${h} \rho {g}=\frac{2 {s}}{{r}} \cos \theta$

$(iii)$ ${RHS} \Rightarrow \varepsilon \times \frac{1}{4 \pi \varepsilon_{0}} \frac{{a}}{{r}^{2}} \times \frac{1}{\varepsilon}=\frac{{q}}{{t}} \times \frac{1}{{r}^{2}}$ $=\frac{{I}}{{L}^{2}}={IL}^{-2}$

$LHS$

${T}=\frac{{I}}{{A}}={IL}^{-2}$

$(iv)$ ${W}=\tau \theta$

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