# Is there an expression for resistance in a circuit derived from Coulomb's Law?

Coulomb's Law states that the repelling force felt by two charges is proportional to the magnitude of their charges and inversely proportional to the square of the distance between them. Defining voltage as the potential energy per unit charge, you get that the voltage is proportional to the charge and inversely proportional to the distance between them.

If instead, substitute the integral of Current with respect to time instead of the charge, then it can be seen that the voltage is proportional to the product of the current and the small time increment (dt), and inversely proportional to the distance between the charges.

Now, I know this is working backwards, however, I remember from a long time ago that the voltage is proportional to the current by the value 'resistance.' So if I apply this to the relationship above, then I find that the resistance is proportional (by the same amount as earlier on) to the time increment (dt) and inversely proportional to the distance between the charges.

Can this be simplified any further? Is there a way of writing resistance similar to this, so that resistance isn't as a function of current or voltage, or have I done all of this completely wrong?

Thanks for submitting your idea! I think "charge" means two different things in your two different equations, and so you can't substitute one for the other. Coulomb's Law comes from electrostatics, where it talks about a single electric charge, and to get the total voltage, you sum over all charges. The integral of current, on the other hand, talks about the total charge moving past a certain point. You can have a current even when the circuit as a whole is electrically neutral. That is, when the negative electrons are balanced by positive ions, so the entire circuit is electrically neutral.

Put another way, the voltage in Coulomb's Law isn't proportional to the integral of the current, or even the current times dt.

The closest thing I can think of to answer this question is the Drude model. In this model, the electrons are accelerated by an (approximately) constant electric field for a short average time $$dt$$, after which it scatters at an atom in the solid structure and, on average, looses all its momentum.

In this model, the resistivity of a metal is (eq. 40) $$\frac{m_e}{e^2 \, n_e \, dt}$$, where $$m_e, e, n_e$$ are the mass, charge and density of electrons.

You can extend this model for time-varying voltages as in the Wikipedia link; or to inhomogeneous electric fields as in the Coulomb law, in which case I believe you would get an inhomogeneous resistivity in the material, but I don't know if someone has tried that.

It is important to remember that resistance is a property of an object (usually a resistor) with physical significance, so one should be careful not to blindly manipulate equations without consideration to the physical things that the variables in them represent. Here, you are combining two equations for voltage, one derived from Coulomb's Law: $$V = \frac{q}{4\pi\epsilon_0 r}$$ and Ohm's Law: $$V = IR$$ The problem is that, while both of these equations give a formula for voltage, they do not describe the same thing physically. The first one describes the electric potential at a point, due to a point charge $$q$$ a distance $$r$$ away, while the second describes the electric potential difference across a resistor of resistance $$R$$ with current $$I$$ flowing through it. Since these equations do not refer to the same thing by $$V$$, they cannot be set equal to one another.

Going back to formulae for resistance, it would not be meaningful to give a formula for it in terms of "distance between the charges," as resistance is not a property of a pair of charges. Instead, resistance describes a physical property of an object, so a formula for it should be given in terms of the material properties of the object.