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, \(\epsilon\) = permittivity, E = electric field, J = current density, L = length.

• A. \(h = \frac{2s \cos\theta}{\rho rg}\)
• B. \(v = \frac{\pi pa^4}{8\eta L}\)
• C. \(W = \Gamma \theta\)
• D. \(J = \epsilon \frac{\partial E}{\partial t}\)

Hint:

Poiseuille's equation specifically defines the rate of flow (\(V/t\)), not the total volume itself.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We check the dimensions of the left-hand side (LHS) and right-hand side (RHS) for each equation. If they don't match, the equation is dimensionally incorrect.
Step 2: Detailed Explanation:
(A) \(h = \frac{2s \cos\theta}{\rho rg}\):
LHS: \([L]\).
RHS: \(\frac{[MT^{-2}]}{[ML^{-3}][L][LT^{-2}]} = \frac{[MT^{-2}]}{[ML^{-1}T^{-2}]} = [L^2]\) --- Wait, \(\rho rg\) is \((ML^{-3})(L)(LT^{-2}) = ML^{-1}T^{-2}\). So RHS is \([MT^{-2}] / [ML^{-1}T^{-2}] = [L]\). Correct.
(B) \(v = \frac{\pi pa^4}{8\eta L}\):
In Poiseuille's law, the term on the RHS has dimensions of volume flow rate (\(L^3 T^{-1}\)).
LHS: \([v] = \text{Volume} = [L^3]\).
RHS: \(\frac{[p][a^4]}{[\eta][L]} = \frac{[ML^{-1}T^{-2}][L^4]}{[ML^{-1}T^{-1}][L]} = \frac{[ML^3T^{-2}]}{[MT^{-1}]} = [L^3T^{-1}]\).
LHS \(\neq\) RHS. Incorrect.
(C) \(W = \Gamma \theta\):
LHS: Work = \([ML^2T^{-2}]\).
RHS: Torque \(\times\) Angle = \([ML^2T^{-2}] \times [1] = [ML^2T^{-2}]\). Correct.
(D) \(J = \epsilon \frac{\partial E}{\partial t}\):
LHS: Current density = \([L^{-2}A]\).
RHS: \([\epsilon] \frac{[E]}{[T]} = [M^{-1}L^{-3}T^4A^2] \frac{[MLT^{-3}A^{-1}]}{[T]} = [L^{-2}A]\). Correct.
Step 3: Final Answer:
Equation (B) is dimensionally incorrect as Volume \(\neq\) Volume flow rate.