Which of the following equations is dimensionally incorrect ? Where t = time, h = height, s = surface tension, θ = angle, ρ = density, a, r = radius, g = acceleration due to gravity, v = volume, p = pressure, W = work done, τ = torque, ∈ = permittivity, E = electric field, J = current density, L = length.
(1) v = \(\frac{\pi pa^4}{8 ηT}\)
(2) h = \(\frac{2 cosθ }{ρrg}\)
(3) J = ∈\(\frac{\partial E}{\partial t}\)
(4) W = τθ
Correct answer: (1) v = \(\frac{\pi pa^4}{8 ηT}\)
Explanation:
v = \(\frac{\pi pa^4}{8 ηT}\) = \(\frac{dv}{dt}\) = Volumetric flow rate (poiseuille’s law)
(ii) hρg = \(\frac{2s}{r}\) cos θ
(iii) RHS ⇒ ε x \(\frac{1}{4 \pi ε_0} \frac{a}{r^2}\) x \(\frac{1}{ε}\)
= \(\frac{q}{t}\) x \(\frac{1}{r^2}\)
= \(\frac{I}{L^2}\) = IL-2
LHS
T = \(\frac{I}{A}\) = IL-2
(iv) W = τθ