Dimensional analysis is a powerful tool in physics and engineering that helps identify relationships between physical quantities by analyzing their dimensions. It is based on the principle that physical equations must be dimensionally consistent. However, there are cases where dimensional analysis might not be as useful or applicable.
Let's consider the given options:
(a) Constant of gravitation (G): Dimensional analysis can be useful in understanding the relationships between various physical quantities, including gravitational forces and masses. While it can help derive relationships like Newton's law of universal gravitation, it might not be as effective in determining the actual value of the gravitational constant (G) itself. The value of G needs to be measured experimentally, and dimensional analysis alone might not provide this constant's precise numerical value.
(b) Constant of proportionality: Dimensional analysis can certainly be useful when dealing with constants of proportionality in equations. It helps establish the relationships between different quantities and determine whether they are consistent with the dimensions of the involved quantities. However, dimensional analysis might not provide the exact numerical value of these constants; experimental measurements are often required for that purpose.
(c) Constant of viscosity: Dimensional analysis is particularly useful in fluid mechanics and can help derive relationships between variables like viscosity, velocity, density, and dimensions of the system. It can lead to dimensionless quantities like Reynolds number, which indicate flow regimes. However, dimensional analysis might not provide the exact numerical value of the viscosity constant; again, experimental data is typically needed for precise numerical values.
(d) Constant of permittivity (ε): Dimensional analysis can play a role in understanding relationships involving permittivity, particularly in the context of electromagnetic phenomena. For instance, the expression for capacitance involves the permittivity of the material between the plates. Similar to the other constants mentioned, dimensional analysis might not yield the exact numerical value of the permittivity constant; experimental measurements are necessary for accurate values.
While dimensional analysis is a valuable tool for understanding relationships between physical quantities, it might not provide exact numerical values for constants in many cases. Experimental measurements are crucial for determining these constants' precise values.