# Why does a voltaic cell induce an electric force proportional to voltage?

I'm trying to understand the intuition behind Ohm's law (that current is proportional to voltage).

I understand the basics of the Drude model: if the electrons frequently collide and on average reset their velocities, then they won't accumulate much momentum and the average velocity will be proportional to the force from the constant electric field $$E.$$

What I'm missing is why we are allowed to substitute the voltage $$V$$ for $$E$$ in the Drude model. In particular, why does the voltage induce an electric field whose strength is proportional to $$V?$$

Normally we start with an electric field, and this naturally induces an electric potential (eg moving an electron against the field increases its potential). But when you start with the purely chemical fact that a reaction could take place and consume electrons at some unknown rate and release free energy in the process, I don't see how this defines the magnitude of an electric field.

My attempt at digging deeper into what the voltage actually means: The voltage of a galvanic cell is $$V = -\Delta G/(nF)$$ where $$n$$ is the number of electrons transferred in the reaction and $$F$$ is Faraday's constant. This makes sense to me: it is roughly the energy released when one Coulomb moves through the conductor and enables the chemical reaction, which matches the definition of voltage (amount of work required per unit of charge to move a test charge from one point to another).

But why is there an electric field proportional to the Gibbs free energy of the reaction? I've seen people informally use language like: "the chemical reaction causes a build-up of electrons at the anode, this is what causes the electric field." Is this the full story? If so, why is the amount of built up electrons at the anode proportional to the Gibbs free energy of the reaction?

Zooming out a bit, it seems strange to me that current should be proportional to Gibbs free energy, because current is the speed of electron flow, and Gibbs free energy is supposed to have nothing to do with the actual rate of the reaction...

4. About the $$V$$ and $$E$$, I will say that the voltage is defined as integral of the electric field; and in a Drude model conductor, there is an external fixed voltage and this define the electric field; the travelling electrons (supplied in arbitrary amount by some source!) adapt their speed to the electric field, giving rise to a current; while overall the conductor is neutral (does not change the electric field, as instead in the galvanic cell).