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

To determine which of the given equations is dimensionally incorrect, we need to analyze the dimensions on both sides of each equation and ensure they match. A dimensionally correct equation will have the same dimensions on both sides.

1. Analyzing \( h = \frac{2s \cos\theta}{\rho rg} \):
   • Left side: [h] = [L] (height is a length)
   • Right side:
     • \( s \) (surface tension) has dimensions [MT^{-2}]
     • \(\rho\) (density) has dimensions [ML^{-3}]
     • r (radius) has dimensions [L]
     • g (acceleration due to gravity) has dimensions [LT^{-2}]
     • Combining: [\frac{2s \cos\theta}{\rho rg}] = \frac{[MT^{-2}]}{[ML^{-3}][L][LT^{-2}]} = [L]
   • Both sides have dimensions of [L], hence it is dimensionally correct.
2. Analyzing \( v = \frac{\pi pa^4}{8\eta L} \):
   • Left side: [v] = [L^3] (volume)
   • Right side:
     • \( p \) (pressure) has dimensions [ML^{-1}T^{-2}]
     • \( a \) (radius) has dimensions [L], so \( a^4 \) has dimensions [L^4]
     • \( \eta \) (viscosity) has dimensions [ML^{-1}T^{-1}]
     • \( L \) (length) has dimensions [L]
     • Combining: [\frac{\pi pa^4}{8\eta L}] = \frac{[ML^{-1}T^{-2}][L^4]}{[ML^{-1}T^{-1}][L]} = [L^2T^{-1}]
   • Left side is [L^3] and right side is [L^2T^{-1}]; thus, this equation is dimensionally incorrect.
3. Analyzing \( W = \Gamma \theta \):
   • Left side: [W] = [ML^2T^{-2}] (work done)
   • Right side:
     • \(\Gamma\) (torque) has dimensions [ML^2T^{-2}]
     • \(\theta\) (angle) is dimensionless
   • Both sides have dimensions of [ML^2T^{-2}], hence it is dimensionally correct.
4. Analyzing \( J = \epsilon \frac{\partial E}{\partial t} \):
   • Left side: [J] = [ATL^{-2}] (current density)
   • Right side:
     • \(\epsilon\) (permittivity) has dimensions [ML^{-3}T^2A^{-2}]
     • \(\frac{\partial E}{\partial t}\) has dimensions [MLT^{-3}A^{-1}]
     • Combining: [\epsilon \frac{\partial E}{\partial t}] = [ML^{-3}T^2A^{-2}][MLT^{-3}A^{-1}] = [ATL^{-2}]
   • Both sides have dimensions of [ATL^{-2}], hence it is dimensionally correct.

Upon analysis, the equation v = \frac{\pi pa^4}{8\eta L} is dimensionally incorrect. Thus, it is the correct answer to the question.