Electron movement in a wire: some crude numbers

Let us have a 1 cm thick and 1 meter long wire made of a metal. Its weight is roughly 1 kg, and it contains some 10²⁵ atoms.

There are

       ~ 10²⁵ conducting electrons

in the wire. The mass of these electrons is

       ~ 10⁻⁵ kilograms.

The charge of these electrons is

       ~ 10⁶ coulombs.

We want to create a 1 ampere current in the wire.

https://en.wikipedia.org/wiki/Inductance

The inductance of the wire is

       L ~ 1 microhenry.

The voltage to increase the current I in the wire is

       V = L dI / dt.

We put a voltage V of 1 microvolt over the wire for 1 second. After that time, the current is 1 ampere.

The Coulomb force of 1 microvolt / meter on the charge 10⁶ coulombs is

        F ~ 1 newton.

The energy of the magnetic field after the operation is

       1/2 L I² ~ 1 microjoule.

The "impulse" which the voltage exerted on the conducting electrons was

       p = 1 newton second.

The collective velocity of the electrons is only

       v = 1 micrometer / second.

It is as if the "inertial mass" of the electrons were a whopping

       10⁶ kilograms.

The kinetic energy and the momentum calculated from the rest mass of the electrons is very small. The momentum is only

       10⁻¹¹ newton seconds,

and the kinetic energy is

       10⁻¹⁷ joules.

Radiation pressure at 4 K is negligible on an object moving 1 micrometer per second

Suppose that we have an object carrying those one million coulombs of charge and moving at v = 1 micrometer per second. That corresponds to a current of one ampere.

We want to calculate the frictional force that the reflection of black body radiation at 4 kelvins imposes on the object.

Let us assume that the area A of the object is one square meter. The power of black body radiation is

       P = A σ T⁴,

where σ = 5.67 * 10⁻⁸ W/(m² K⁴) is the Stefan-Boltzmann constant and T is the temperature. We have

       P = 1.4 * 10⁻⁵ W

at 4 K. The radiation pressure force by the power P is

       P / c = 5 * 10⁻¹⁴ N.

The pressure is almost the same on the each side of the object, except for the Doppler shift caused by the tiny velocity v = 1 μm/s.

The effect of the Doppler shift is

       4 v / c,

because the reflection adds a factor of two, and the effect is on both sides of the object.

We conclude that the frictional force on the object by black body radiation is

       F ~ 4 v / c * P / c

           = 7 * 10⁻²⁸ N.

We calculated above that, to create the magnetic field for a 1 ampere current in a one meter wire, we have to expend 1 newton second of impulse. The force F is negligible relative to that.

The friction from black body radiation is negligible at 4 K. The resistivity at a low temperature has to come from collisions with the lattice. Those collisions move a large amount of impulse from the lattice to the electrons.