Self-induction and drude equation

[dirad]

why are the effects of self-induction not taken into account when writing electron transport equations in conductors under effect of emf such as with drude equation?

 

[Baluncore]

The Drude equations apply a statistical model to point charges distributed within a conductor.
The boundaries of a conductor constrain the movement of those charge carriers.
Self-inductance is a parameter applied externally to the geometrical shape of the conductor.

The magnetic forces between charges moving due to the Drude applied EMF are perpendicular to the electric forces due to the same applied EMF. Those magnetic forces will ensure that the electrons remain distributed throughout the section of the conductor, which is an assumption of the Drude model.

 

[dirad]

current lags behind emf

can you please clarify the relation between these magnetic forces and drude equation?

 

[Baluncore]

There is no relation between magnetic forces and the Drude model.
The Drude model applies to the inside of a very short conductor.
Self inductance applies to the outside of a longer conductor.
They are independent concepts.

 

[dirad]

switching an electric current on

so, what exactly happens in an electric circuit when a current is switched on from the microscopic point of view?

 

[Baluncore]

In effect Maxwell's equations, constrained by the boundary conditions of the physical system apply.
There you have it. How you visualise what happens at and following a turn on transient is really determined by why you are modelling the system. Zen and the art of visualising the electro-magnetic circuit requires a qualitative approach. The English language is a poor substitute for mathematics, so from here on you can expect non-mathematical gross generalisations on the microscopic scale.

Since the currents on the surface skin of the conductor determine the currents within the conductor, we are best leaving them until later by which time it will be realized why bulk things happen so much slower deep inside a conductor than on the surface.

The key to understanding the circuit model is in recognising the importance of the surface. That is where the external model will meet and interact with the internal model.

First we will divert to consider a magnetic field cutting the surface of a conductor. That will generate a microscopic current flowing along the surface perpendicular (90°) to the field. The induced surface current will itself generate a magnetic field, again perpendicular (90° + 90° = 180° = backwards) and so opposing the original incident magnetic field. In effect these two fields cancel in the surface of a good conductor and so prevent the field from significantly penetrating the surface. The result is that, to an EM wave, the surface of a conductor looks like a mirror. A super-conductor is a perfect mirror, a good conductor is a pretty good mirror and a resistor is a poor mirror.

Now back to the circuit. If we look at the arrangement of the surface of the conductors then, initially there is a static electric field determined by the EMF sources provided, it is shaped by the equipotential boundaries of the conductor geometry in space.

We will start on the surface of the open switch contacts with a differential EMF. At the instant the two conductors make contact the differential EMF must disappear at that point of contact.

The initial static electric field is disturbed beginning at the instant of switch closure. Over time it will rearrange itself to the new electrostatic configuration with the addition of a magnetic field generated (or supported) by the current that begins to flow in the surface of the conductors.

A surface EM wave starts to radiate from the point of switch closure at close to the speed of light. That wave can be modeled as hugging it's reflection in the surface of the conductor and spreading until it cancels itself traveling in the opposite direction. I say “close” to the speed of light because the dielectric constant of an insulation layer, maybe only a very thin coating of oxide will very slightly delay the propagation.

At every point during the expansion of the surface wave there will be radiated smaller spherical EM waves that encounter and link to the conductive surface of the other conductors in the circuit. The interaction or coupling of all these radiated waves gives the circuit the property we call self inductance. Now let us look at the surface from the inside. The surface current is constrained by the surface and so must be flowing parallel with the surface, (or it would be emitted, which would change the assumed circuit). It is therefore not possible for the current to simply flow into the depths of the conductor. But there is a way of getting deeper currents to flow. Where the magnetic field in one layer is not fully canceled by the reflecting current, there will be some residual field. That field will cause a deeper current to flow and so on until a current is flowing throughout the cross-section of the conductor. At no point does any current flow perpendicular to the surface. Available deep charge carriers begin to move axially. This movement is longitudinal, perpendicular to the voltage gradient.

The start of longitudinal current flow will penetrate at the speed of light, but the current will not build up for quite some time. The delay is a function of conductivity and magnetic permittivity. Doing the numbers shows that in copper for example, the bulk of current flow penetrates at about 5 metres per second. This phenomena, called the “skin effect”, explains why currents at high frequency only flow in the surface of the conductor. DC current on the other hand has plenty of time to soak magnetically deep into the conductor.

Because less cross-section of the conductor is available to high frequency current, the total resistance appears to be proportionally higher for RF. Since at RF almost no current flows inside a solid bar, inductors can be modeled and made from copper tube. (That also makes it easier to liquid cool them). Another adaption is to make RF connections with a flat tape having a thickness twice the skin depth. That will not only minimise the mass of the conductor but it will also minimise the self inductance of the connection by keeping parallel currents apart, and so less coupled. Now that we have skimmed the surface of understanding circuit transients, you will find that there is plenty to study in depth. The current flowing inside the slightly resistive conductors will suffer an EMF drop along the conductor and a some version of the Drude model can be applied.

The Drude model will never become a tool for circuit modelling. It will always remain a statistical model that can be used to explain the observed phenomena of current flow in conductors.

 

[dirad]

your reply was extremely helpfull

your reply was extremely helpful. i understand that Maxwell's equations in material media will be the basis of explanation. emphasis on role of surface and boundaries was illuminating. i began reading about skin effect and suchlike.

my own thinking usually revolves about DC circuits containing chemical source for EMF attached to a wire having resistance and inductance, trying to visualize the static conditions prior to starting the current and the transient behaviour of electron currents. my understanding was that there is some kind of lag of the current behind the value obtained in Ohm's law, trying to visualize this as some kind of electron inertia plus back EMF.

i know that these are rough approximations to Maxwell's equations, but simple models might be helpful to begin with.