Alternating current distribution is non-uniform in real conductors with finite dimensions and rectangular or circular cross-sections. This is because the alternating current flow creates eddy currents in real conductors leading to current crowding, following Faraday’s laws.

AC currents, being time-varying, produce non-uniform distributions across the cross-sectional area of a conductor. The conductor offers a high-frequency resistance, and for the approximation, we can assume the current to flow uniformly in the conductor, in a layer one skin deep, just below the surface. This phenomenon is known as the skin effect. However, this is only a simple explanation, with the actual distribution of current being much more nuanced, even within an isolated conductor.

For instance, what is the current distribution within a cylindrical conductor with a diameter 2.5 times greater than the skin depth at the frequency of interest? For the answer, it may be necessary to look closely at the physics of skin effect, and the way skin depth is typically derived.

Skin effect is caused by a basic electromagnetic situation. This is related to the propagation of electromagnetic waves inside a good conductor. Textbooks typically examine the propagation of a plane wave within a conducting half-space.

Euclidean space is typically three-dimensional, consisting of length, breadth, and height. A plane can divide this space into two parts, with each part being a half-space. Therefore, a line, connecting one point in one half-space to another point in the other half-space, will intersect the dividing plane. Plane waves propagate along the dividing plane in the conducting half-space.

Now, plane waves consist of magnetic and electric fields that are perpendicular to the direction of propagation and each other. That is why these waves are also known as transverse electromagnetic or TEM waves. Moreover, within a plane wave, all points on planes perpendicular to the direction of propagation, experience the same electric and magnetic fields.

For instance, considering the electric field (E) is in the z-direction, the magnetic field (H) will be in the x-direction, while the wave propagates in the y-direction. Therefore, assuming a plane wave propagation, the electric and magnetic fields remain constant along the x or y direction, and change only as a function of y.

Moreover, for a good conductor, the electric field and the current density are interrelated to the conductivity of the conductor. Using these parameters allows us to calculate the current density, and the skin depth, by solving Maxwell’s equation.

Maxwell’s equation tells us that the amplitude of the current density at the skin depth decreases at the surface of the conductor. It also gives an initial idea of the change in current density at any instant in time as we go deeper into the conductor.

The equation allows us to relate the skin depth to the wavelength within the conductor. The attenuation constant and phase constant of a good conductor are inversely related to the skin depth. It is easy to see that a single wavelength within the conductor is about 6 times larger than the skin depth. This also means the current density will attenuate significantly at a distance of one wavelength.