Dimensions of Newton's Law of Gravitation and Coloumb's Law

[Pinu7]

As we know, Newton's Law of Gravitation is
$$\[ {\mathbf{F}} = \frac{{Gm_1 m_2 }} {{r^2 }} \]$$
and Coulomb's law is

$$\[ {\mathbf{F}} = \frac{{Qq_1 q_2 }} {{r^2 }} \]$$

We know from comparing the dimensions of the first equation that G, the gravitational constant, has the dimension
$$\[ [M^{ - 1} L^3 T^{ - 2} ] \]$$

But for Coulomb's law, we assume Q is dimensionless. Why do we make this assumption?

 

[D H]

Where did you get the idea that Q (typically κ, not Q) is unitless?

Dimensional analysis tells you what the units for κ must be:

$$d(\kappa)=d(F)*d(r)^2/d(q)^2 = \frac{FL^2}{Q^2} = \frac{ML^3}{T^2Q^2}$$

where the d() extracts the dimensions of the quantity in question. This is not a dimensionless quantity. The only way it can be dimensionless is if charge is expressed in non-rational units, $$d(q) = \sqrt{ML^3}/T$$

See http://scienceworld.wolfram.com/physics/CoulombsConstant.html

 

[astrokreng]

The assumption that Q is dimensionless in Coulomb's law is based on experimental evidence and theoretical considerations. In most cases, Coulomb's law is applied to point charges, which have no physical dimensions and therefore do not have any inherent charge density. Additionally, the units of charge, coulombs, are defined as the amount of charge that passes a point in one second when one ampere of current is flowing. This definition does not include any physical dimensions, further supporting the assumption that Q is dimensionless. Moreover, incorporating a dimension for Q would introduce additional complexity and potential errors in calculations, without providing any additional physical insight or predictive power. Therefore, for simplicity and consistency with experimental observations and theoretical considerations, we assume that Q is dimensionless in Coulomb's law.