Magnetic Fields and Currents

[LucasGB]

Suppose we are looking at the cross-section of a cylindrical current-carrying wire. Due to the motion of the electrons in the wire, it can be said that at all points at the surface of the wire a magnetic field tangential to the wire exists. Since there are electrons moving in that region as well, a radial Lorentz force arises on these electrons.

The questions:
1. Is this analysis correct? Does this force exists?
2. And if it does, wouldn't this cause all the electrons in the wire to pile up at its center?

This is fundamentally a problem concerning the effect of a wire's magnetic field on itself.

 

[Bob S]

These are very good questions. First, in a charged-particle beam like protons for example, there is both a radial repulsive Coulomb force pushing the beam apart, and a radial attractive magnetic force pulling the beam to the center. The Coulomb force is larger than the magnetic force at all velocities except at extreme relativistic velocitires, when the two opposing forces are equal.

In a wire there is no Coulomb force due to the equality of + and - charge density in the wire even when there is a current, but the finite wire resistance (I.e., IR drop) causes the current to be uniform over the cross section of the wire. Furthermore, the velocities of individual electrons in a current-carrying wire are very low. So although there is a radial Lorentz v x B force, it is very small.

Look at it another way. Suppose every copper atom had one electron in the conduction band. So there would be 1 mole of conduction electrons per 63 grams, and with a density of 9 grams per cm³, there would be ~0.14 moles (~13,000 Coulombs**) of conduction electrons per cm³. If these conduction electrons were moving at the speed of light, the current would be ~4 x 10¹⁴ amps per cm². In actuality, hollow water-cooled copper bus bar is limited to about 1000 amps per cm². The implication here is that the average electron velocity in a copper conductor is ~0.1 cm/sec. Since the Lorentz force is proportional to velocity, the force is negligible.

I hope this helps.

** 1 mole of electrons is 96,485 Coulombs.

Bob S

 

[LucasGB]

Thank you for the very good reply. I hadn't taken into consideration how slow electrons normally move. I must point out, though, that even though the Lorentz force is very small, it is as you said, the only force acting on the electrons. Wouldn't this mean, after all, that they should pile up at the center?

Perhaps, as they start doing so, Coulomb's force becomes different from zero and balances Lorentz's force. Perhaps in all current-carrying wires there is a balance between Coulomb's and Lorentz's force keeping electrons in place. What do you think?

 

[Born2bwire]

But you still have to contend with the implications that arise from regions of net charge. If the electrons are forced towards the interior of the wire, then they will leave behind positively charged atoms in the exterior lattice. In addition, by grouping up in the interior, the electrons will create a region of net negative charge. Both the positive charge on the surface and the negative charge in the interior will work to move the electrons back out, negating the Lorentz force.

 

[Per Oni]

Thank you for the very good reply. I hadn't taken into consideration how slow electrons normally move. I must point out, though, that even though the Lorentz force is very small, it is as you said, the only force acting on the electrons. Wouldn't this mean, after all, that they should pile up at the center?

Perhaps, as they start doing so, Coulomb's force becomes different from zero and balances Lorentz's force. Perhaps in all current-carrying wires there is a balance between Coulomb's and Lorentz's force keeping electrons in place. What do you think?

From a relativity point of view, for conduction electrons in a rest frame the +ve lattice is moving and is therefore length contracted. Conduction electrons along the outside of the wire are therefore attracted inwards but those at the centre find that they are being pulled in all radial directions and therefore for them this force cancels out. Net result of all conduction electrons is still inwards but is again cancelled, as you mention, by coulomb forces.

On the other hand, in case of a high frequency current, the conduction electrons are forced to run towards the outside of the wire, and the wire can be made hollow without increasing resistance significantly.

Another point to bear in mind is this: the average speed of conduction electrons in the direction of the current is very slow but they have individual speeds of up to (+ and -) 10^6 m/s and therefore some will experience much higher forces.

 

[Bob S]

Thank you for the very good reply. I hadn't taken into consideration how slow electrons normally move. I must point out, though, that even though the Lorentz force is very small, it is as you said, the only force acting on the electrons. Wouldn't this mean, after all, that they should pile up at the center?

Suppose all the current is in the center of a resistive wire. Then there is an IR voltage drop along the center of the wire. But if there is no current along the surface of the wire, there is no voltage drop along the surface. But Kirchhoff's law requires no net loop voltage drop around the loop. From symmetry, the voltage drop along surface must equal voltage drop along center. So there must be a surface current.

Bob S