I have to check that Maxwell's equations are dimensionally correct.
How the hell can I do this?
Thank you very much.
Oh my, I have to get my old MIT T shirt out of the closet?
well, the amount of E coming out is proportional to charge inside for example. That is one of them.
It can be expressed in integral or in differential form.
In integral for example:
integral over closed surface of E dot DA = Q/eo
E has units of Newtons /Coulomb or force per unit charge
so on the left you have
Newtons * meters^2 /Coulombs
==================
Now what on the right?
Q is Coulombs obviously
and eo is in Coulombs^2/(Newton meter^2)
see for example https://www.britannica.com/science/permittivity
so on the right you have
Coulombs * Newton *meter^2/Coulombs^2 = Newtons meter^2/Coulombs
which is luckily what you had on the left
Just keep going for the others