Magnetic field inside conductor and linearity of eddy currents

Eddy currents are currents that are generated in a conductor to produce magnetic fields that oppose the magnetic field originally produced by current flowing in the conductor. The figure below (from Wikipedia's skin effect article) depicts this nicely:

Current $I$ flows up through a wire, producing a counterclockwise magnetic field $H$ into and out of the page. By Lenz's law, to oppose this change in flux, an eddy current $I_W$ is created that flows in concentric circles centered along $H$ which produces a magnetic field that opposes $H$. This is known as the skin effect.

Questions:

1. Is the skin effect worse for alternating currents (AC) than for direct currents (DC) since for DC, the change in magnetic flux goes to zero after the initial transient period when current $I$ starts to flow?
2. When the eddy currents are flowing, is the magnetic field inside the conductor zero?
3. If the answer to 2 is no, then are eddy currents linear? I.e. for a given location $(r, \theta, z)$ inside the conductor, does doubling the current result in a doubling of the magnetic field? Or does the presence of eddy currents diminish / enhance the magnetic field in a nonlinear way?

1. Is the skin effect worse for alternating currents (AC) than for direct currents (DC) since for DC, the change in magnetic flux goes to zero after the initial transient period when current starts to flow?

Yes, you are right. The skin effect will vanish for DC very quickly since there is no changing current (except from turning on or off) and hence no changing magnetic field.

1. When the eddy currents are flowing, is the magnetic field inside the conductor zero?

No, not exactly. The current density decreases exponentially from the surface by $$J=J_S\cdot\mathrm{e}^{- d /\delta}$$ where $J_S$ is the current density at the surface, $d$ the depth from the surface and $\delta$ the skin depth which is the depth at which the current density has fallen to $J_S/\mathrm{e}$. This means over 98 % of the current flows between the surface and $4\delta$ depth. (Source: https://en.m.wikipedia.org/wiki/Skin_effect). Therefore there is only a minor current flowing in the middle of the conductor which, I think, can be neglected in most applications. But nonetheless there will be a small magnetic field due to the small current flowing.

1. If the answer to 2 is no, then are eddy currents linear? I.e. for a given location $(r,θ,z)$ inside the conductor, does doubling the current result in a doubling of the magnetic field? Or does the presence of eddy currents diminish / enhance the magnetic field in a nonlinear way?

As we can see from the above equation doubling the current density at the surface yields indeed a doubling of the current density inside (the skin depth $\delta$ does not depend on the current).

Intuitively a doubled current leads to a doubled magnetic field. This leads also to a doubled magnetic field change resulting in a doubled back EMF (induced electric field). This gives doubled eddy currents. But the eddy currents induce an opposing magnetic field (on the inner side) counteracting the former magnetic field. In other words the current decreases by the eddy currents resulting in a smaller magnetic field leading to smaller eddy currents. So you were right.

• Thank you, @EuklidAlexandria. Could you please clarify your second-to-last sentence? I'm confused about the part where the eddy currents themselves get reduced. So the doubling of eddy currents eventually leads to a decrease in the eddy currents? Is there some kind of equilibrium that is reached between the input current $I$ and the eddy current $I_w$? Said another way, if the initial voltage across the conductor is $V$ and you double it to $2V$, by what factor does $I$ (and the magnetic field $H$ induced by it) change after accounting for $I_w$ (after the equilibrium has been reached)? – Vivek Subramanian Aug 21 '18 at 0:07
• Any thoughts, @EuklidAlexandria (or anyone else)? – Vivek Subramanian Aug 26 '18 at 4:15
• @VivekSubramanian The eddy currents $I_w$ oppose the current $I$ leading to smaller net current which results in a smaller magnetic field and eventually to smaller eddy currents. Since the skin depth $\delta$ does not depend on the applied voltage, twice as much current will flow if the voltage is doubled according to Ohm's law. For a detailed derivation see this. – EuklidAlexandria Sep 8 '18 at 18:37
• @VivekSubramanian Note also that the current density inside is phase shifted (but not by $90\deg$, since there is resistance) as mentioned in the above link. Furthermore, since energy gets transported from the outside to the wire through EM-wave (transportation of AC voltage), the skin depth characterizes how far waves propagate into the wire. – EuklidAlexandria Sep 8 '18 at 18:46