Using the base units of each physical quantity

To prove that an equation is homogeneous, we must show that the dimensions of the Left Hand Side (LHS) are equal to the dimensions of the Right Hand Side (RHS) using base SI units.

First, let's clarify the terms in the equation $F = \frac{\mu_0 I_1 I_2}{2a}$:

• $F$: Force (as stated by you).
• $\mu_0$: Permeability of free space (not permittivity, which is $\epsilon_0$).
• $I_1, I_2$: Electric current.
• $a$: Distance.
• 2: A dimensionless constant.

Step 1: Determine the base units for the Left Hand Side (LHS).
The LHS is Force ($F$).
Force is defined as mass $\times$ acceleration ($F = ma$).
The base units for mass (m) are kilograms (kg).
The base units for acceleration (a) are meters per second squared (m/s$^2$).
So, the base units for Force ($F$) are:

$$$$