JEE MAIN - Physics (2021 - 31st August Morning Shift - No. 14)
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, T = torque, $$\in$$ = permittivity, E = electric field, J = current density, L = length.
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, T = torque, $$\in$$ = permittivity, E = electric field, J = current density, L = length.
$$v = {{\pi p{a^4}} \over {8\eta L}}$$
$$h = {{2s\cos \theta } \over {\rho rg}}$$
$$J = \in {{\partial E} \over {\partial t}}$$
$$W = \Gamma \theta $$
Explanation
(a) $${{\pi p{a^4}} \over {8\eta L}} = {{dv} \over {dt}}$$ = Volumetric flow rate (Poiseuille's law)
(b) $$h\rho g = {{2s} \over r}\cos \theta $$
(c) $$\varepsilon \times {1 \over {4\pi {\varepsilon _0}}}{a \over {{r^2}}} \times {1 \over \varepsilon } = {q \over t} \times {1 \over {{r^2}}}$$
$$ = {1 \over {{L^2}}} = I{L^{ - 2}}$$
LHS
$$T = {I \over A} = I{L^{ - 2}}$$
(d) W = $$\tau$$$$\theta$$
Option (a)
(b) $$h\rho g = {{2s} \over r}\cos \theta $$
(c) $$\varepsilon \times {1 \over {4\pi {\varepsilon _0}}}{a \over {{r^2}}} \times {1 \over \varepsilon } = {q \over t} \times {1 \over {{r^2}}}$$
$$ = {1 \over {{L^2}}} = I{L^{ - 2}}$$
LHS
$$T = {I \over A} = I{L^{ - 2}}$$
(d) W = $$\tau$$$$\theta$$
Option (a)
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