Q1Electrical Materials

(a) Why skin effect is absent in DC fields: skin effect is caused by electromagnetic induction — the alternating magnetic flux linked with the current induces eddy EMFs inside the conductor that oppose current flow most strongly at the center, forcing current toward the surface. Induction requires a time-varying flux (Faraday's law, EMF ∝ dΦ/dt). For direct current the flux is constant, dΦ/dt = 0, no internal opposing EMFs are induced, and the current therefore distributes itself uniformly over the full cross-section: skin effect simply cannot arise in a steady DC field.

(b) Derivation of skin depth: inside a good conductor, Maxwell's equations with Ohm's law (J = σE, displacement current negligible compared to conduction current) combine to give the diffusion equation for the field/current density. For a sinusoidal field of angular frequency ω penetrating in the x-direction:

whose decaying solution is E(x) = E₀e^{-x/δ}e^{-jx/δ}, i.e., magnitude decaying exponentially with depth, with:

This is the required relation: skin depth is inversely proportional to the square root of frequency f, permeability μ, and conductivity σ. At depth δ the current density has fallen to 1/e (≈37%) of its surface value; almost all the current flows within a few skin depths of the surface. The relation explains the practical consequences of skin effect — higher frequency, higher permeability (e.g., iron), or higher conductivity all shrink δ, worsening the confinement of current to the surface, which motivates stranded/litz conductors, hollow tubular bus-bars and the use of thin laminations at power frequencies and above.

(c) Numerical evaluation: with σ = 10 S/m (mho/m), f = 10 MHz = 10⁷ Hz, μ = μ₀ = 4π×10⁻⁷ H/m:

The skin depth is approximately 5.03 cm. This relatively large value (compared to the ~21 μm skin depth of copper at the same frequency) reflects the very low conductivity of the given material — 10 S/m is closer to a lossy medium such as sea water than to a metal — directly illustrating the inverse-square-root dependence of δ on σ derived in part (b): reducing conductivity by a factor of ~5×10⁶ relative to copper increases skin depth by a factor of over 2000.

Practical implication of the frequency dependence: because δ ∝ 1/√f, the same conductor exhibits sharply different AC behavior across the frequency spectrum — at 50/60 Hz power frequency, copper's skin depth is around 9 mm, so ordinary solid conductors up to roughly 20 mm diameter still carry current fairly uniformly and skin effect is a secondary correction; at radio and microwave frequencies (MHz to GHz, as in this problem), δ shrinks to micrometres in good conductors, so essentially all current flows in an extremely thin surface layer, and RF conductor design (silver-plated waveguide walls, PCB trace surface finish, coaxial cable shields) is governed almost entirely by surface conductivity and roughness rather than by the bulk material deeper inside — a direct engineering consequence of the δ ∝ 1/√(fμσ) relation derived in this question.