I want to understand how does the energy transfer from the battery to the resistor in a simple DC circuit. I read that it is due to the surface charge the battery creates on the wire.
So why is this charge a surface charge not volumetric one? And if it's on the surface why is there an electric field inside the wire?
I'd be happy with your help for me to understand this confusing topic.
So why this charge is surface charge not volumetric one.
The reason that it is a surface charge and not a volumetric charge is because a surface charge causes a sudden change in the E-field direction from outside to inside the wire.
Inside the wire the E-field points along the wire. This is necessary due to Ohm's law and the fact that current flows along the wire. So the E-field and the current point in the same direction. However, by Poynting's theorem, the flow of energy is perpendicular to both the E-field and the B-field. Inside the wire the B-field is circumferential and the E-field is along the axis of the wire, so the energy flow is radially into the wire. This power flux inside the wire is equal to the power dissipated by resistive heating of the wire itself.
Outside the wire the E-field points predominantly radially into or out of the wire. So outside the wire the B-field is still circumferential and with a radial E-field we get power flowing longitudinally along the wire from the source to the load.
Thus, we have a situation where the E-field changes direction suddenly at the surface of the wire. It goes from mostly radial outside the wire to longitudinal inside the wire. The surface charge is what accomplishes that sudden change in the direction of the E-field. A volumetric charge density would provide for a gradual change in the E-field, not the needed sharp change. That is essential for understanding power flow.
In a simple DC circuits the charge carriers will drift through the bulk volume of the wire and resistor.
Collisions in the resistor (and also wires) will be converted in heat and effectively transferring battery energy to the resistor and less so to the wire.
There will be net charge accumulation on the surface of the wire and resistor maintaining the electric field through it. This field is present in their interior volume continuously accelerating the charge carriers until they collide (very rapidly actually).
See e.g. does-electricity-flow-on-the-surface-of-a-wire-or-in-the-interior and do-electrons-flow-only-on-the-surface-of-a-wire.
Charges of equal sign repel each other. In conductor, this makes any non-zero charge imbalance to spread out and decrease in magnitude. The stable state in electrostatics and in constant dynamic current (DC) circuits is that all excess charge is on surface of the conductor, the charged particles cannot get farther from each other, because (in common low voltage and low temperature cases) to leave the conductor they would have to overcome quite a high energy barrier. So they stay on the surface.
Electric field in case of DC circuits is given by the Coulomb formula. Since there is current flowing in the conductor, there has to be some electromotive force that maintains it. In ordinary wire in DC circuit the only such force can be the Coulomb electric force of charges in the system.
