It is helpful to understand the basic functioning of a coaxial cable. But first there are two important phenomenons that must be understood in order to proceed.
Skin Effect
When direct current (time invariant current) passes through a conductor it tends to uniformly use the entire cross sectional area of the conductor. When alternating current (time variant current) passes through a conducting material, it has a tendency to concentrate the current near the surfaces of the conductor. The occurs due to the changing direction of the magnetic field associated with the alternating current resulting in a back electromotive force that tends to push the conducting electrons towards the surface of the conducting material. As the frequency of the alternating current is increased, the crowding toward the surface becomes more pronounced. Since there is less conducting area, the resistance rises with frequency.
In a common conductor such as copper, the skin effect can be quite pronounced at typical amateur radio frequencies. For example, at 1 MHz, 63% of the current flows in the outer ~0.067 mm of a round copper wire. The current decreases exponentially and this 63% level is considered one skin depth. 98% of the current is contained in four skin depths (0.298 mm) from the surface of the wire. Past five skin depths, there is for all practical purposes no current flowing in the wire. Thus the center of the wire with a radius greater than five skin depths is essentially wasted material and can be removed (making a tubular shape) or replaced with other structural materials. We see the latter in amateur radio circles in the CopperWeld® brand of wire where the inner material is steel (for strength purposes) but with an outer copper coating (for RF purposes). If the thickness of the copper coating is at least 5 skin depths for the frequency in use, the RF current will never "see" the higher resistance steel core due to the skin effect.
Proximity Effect
When alternating current flows through a conductor and this conductor is in close proximity to another conductor carrying alternating current, the current in the conductors will tend to be crowded into smaller regions of the conductor resulting in increased resistance. It is important to note that this is in addition to the skin effect. Thus the proximity effect tends to further increase the effective resistance of the conductor beyond that of the skin effect alone.
If the two wires are carrying current in opposite directions, the crowding tends to occur on the surfaces of the wire that are closer together. If the current is flowing in the same direction, the crowding tends to occur on the surfaces that are further apart.
Coax Cable
Now consider a coax cable with an outer shield that is several skin depths thick and with an alternating current flowing only on the inner (center) conductor. The shield is completely disconnected for this description. The alternating current flowing in the center conductor develops a magnetic field around it. Since the outer shield conductor is several skin depths thick, there is practically no magnetic field developed within the outer conductor.
We can apply Ampere's Law to determine that the magnetic field of the inner conductor will induce an equal and opposite current on the inside of the outer shield (this is a working example the proximity effect). The current on the inside of the outer conductor must close on itself so that the current flowing on the outside of the shield is the same as the current flowing on the inside of the shield.
Now if we connect the shield as it would be used in the usual condition for an amateur radio application, we have current flowing on the outside of the shield (due to skin effect) that is fully cancelled out due to the opposite induced current from the inner conductor described earlier. This leaves us with only the current on the inside of the outer conductor. It should be clear from this description that the result of the current flowing only on the inside of the outer conductor is therefore a result of both the skin effect and the proximity effect.
The net result is that the shield of the coax cable acts as an electrostatic shield that, for all practical purposes, does not allow the electromagnetic energy to radiate from the coax cable.
Common Mode Current on Coaxial Cable
If we now bring another conductor carrying an alternating current (such as an antenna element) in close proximity to the coax, it will induce a current in the shield. If the shield is several skin depths thick, none of this induced current will appear on the inside of the shield conductor (or anywhere else inside the coaxial cable) but rather it will flow on the outside of the shield due to skin effect.
In this condition, we have two unique currents flowing on the shield of the coax – the current flowing on the inside of the shield that is generated by the transmitter connected to the coax as well as the current that is flowing on the outside of the shield due to magnetic coupling from the antenna. With sufficient thickness of the shield, these two currents do not “see” each other since neither will (effectively) pass through the center of the shield to interact with the other. This is quite different from say the reflected current that occurs due to a mismatched load where the forward and reflected currents do “see” each other and interact to create a standing current wave.
Since the two currents have no interaction along the length of the coax it is, for all practical purposes, as if they are traveling on separate wires. Thus when describing coaxial common mode current, the exterior of the shield is often described as the “third wire” of the coaxial cable. Of course this is simply a simile but it closely fits the observable characteristics.
