Dimensional Analysis

From Coulomb?? law, the force of attraction/repulsion between two point charges

q

q

separated by distance

r

F%20%3D%20%5Cfrac%7B1%7D%7B4%5C%2C%5Cpi%5C%2C%5Cvarepsilon_%7B0%7D%7D%20%5Cfrac%7Bq%5E%7B2%7D%7D%7Br%5E%7B2%7D%7D

%5CRightarrow%20%5Cvarepsilon%20_%7B0%7D%20%3D%20%5Cfrac%7B1%7D%7B4%5C%2C%5Cpi%20%7D%20%5Ccdot%20%5Cfrac%7Bq%5E%7B2%7D%7D%7BFr%5E%7B2%7D%7D

where

e_%7B0%7D

is electric permittivity.
Dimensions of

%5Cvarepsilon_%7B0%7D%20%3D%5Cfrac%7B%5Cleft%5BAT%5Cright%5D%5E%7B2%7D%7D%7B%5Cleft%5BMLT%5E%7B-2%7D%5Cright%5D%5Cleft%5BL%5E%7B2%7D%5Cright%5D%7D

%5Cvarepsilon_%7B0%7D%20%3D%20%5Cleft%5BA%5E%7B2%7D%5C%2CM%5E%7B-1%7D%5C%2CL%5E%7B-3%7D%5C%2CT%5E%7B4%7D%5Cright%5D

As, the formula of Reynold?? number

R_%7Be%7D%20%3D%20%5Cfrac%7BV_%7Bc%7D%5Crho%20D%7D%7B%5Ceta%7D

So, the dimension of

R_%7Be%7D%20%3D%20%5Cfrac%7B%5Cleft%5BM%5E%7B0%7DLT%5E%7B-1%7D%5Cright%5D%5Cleft%5BML%5E%7B-3%7DT%5E%7B0%7D%5Cright%5D%5Cleft%5BM%5E%7B0%7DLT%5E%7B0%7D%5Cright%5D%7D%7B%5Cleft%5BML%5E%7B-1%7DT%5E%7B-1%7D%5Cright%5D%7D

%5CRightarrow%20%5Cleft%5BR_%7Be%7D%5Cright%5D%20%3D%20%5Cleft%5BM%5E%7B0%7DL%5E%7B0%7DT%5E%7B0%7D%5Cright%5D

Hence, it is a dimensionless quantity.
Whereas dimension of
(i) Coefficient of viscosity

%3D%20%5BM%5E0L%5E%7B-1%7D%20T%5E%7B-1%7D%20%5D

(ii) Coefficient of friction

%3D%20%5BM%5E0%20L%5E0%20T%5E0%20%5D

(iii) Universal gravitational constant

%3D%20%5BM%5E%7B-1%7D%20L%5E%7B3%7D%20T%5E%7B-2%7D%20%5D

(iv) Velocity of light

%3D%20%5BM%5E0%20LT%5E%7B-1%7D%20%5D

Hence, the dimensional formula of Reynold?? number is same as coefficient of friction.

Chapter Questions
2 MCQs