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

(C) $v=\frac{\pi p a^{4}}{8 \eta L}$

(A) $\mathrm{W}=\Gamma \theta$

(B) $\mathrm{h}=\frac{2 \mathrm{~s} \cos \theta}{\mathrm{prg}}$

(D) $\mathrm{J}=\in \frac{\partial \mathrm{E}}{\partial \mathrm{t}}$

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Units & Measurement

Which of the following equations is dimensionally incorrect ?
Where

t =

time,

h =

height,

s =

surface tension,

θ =

angle,

p =

density,

a, r =

radius,

g =

acceleration due to gravity,

v =

volume,

p =

pressure,

W =

work done,

Γ =

torque,

\in=

permittivity,

E =

electric field,

J =

current density,

L =

length.

Answer: (C)

Work equals torque times angle,

M L^{2} T^{- 2}

on both sides since angle is dimensionless, so that relation is sound. The capillary rise

\frac{2 s cos θ}{ρ r g}

reduces to

\frac{M T ^{- 2}}{( M L ^{- 3} ) ( L ) ( L T ^{- 2} )} = L

, a length, so it is sound. The displacement-current relation

J = ϵ \frac{\partial E}{\partial t}

gives a current density on both sides. In

\frac{π p a ^{4}}{8 η L}

the right side is

\frac{( M L ^{- 1} T ^{- 2} ) ( L ^{4} )}{( M L ^{- 1} T ^{- 1} ) ( L )} = L^{3} T^{- 1}

, a volume flow rate, not a volume

L^{3}

. Equating a volume to

\frac{π p a ^{4}}{8 η L}

is dimensionally incorrect.

JEE Main · 2021 · 31 Aug 2021 · Shift 1 · Single correct option