The question asks about the condition under which the skin effect in an electrical conductor becomes more prominent for an alternating current (AC).

Understanding the Skin Effect Phenomenon

The skin effect is a physical phenomenon observed in electrical conductors carrying alternating current. It describes how the AC tends to flow near the surface (or "skin") of the conductor, rather than being distributed uniformly throughout its entire cross-section. This happens because the alternating magnetic field generated by the AC induces circulating currents within the conductor itself, known as eddy currents. These eddy currents oppose the flow of the main current, particularly in the central regions of the conductor.

Analyzing the Role of Frequency

The prominence of the skin effect is strongly dependent on the frequency of the alternating current.

• Low Frequencies: At very low frequencies, such as the standard power line frequency (50 Hz or 60 Hz), the skin effect is generally minimal. The eddy currents induced are relatively weak, and the AC flows almost uniformly across the conductor's cross-section.
• High Frequencies: As the frequency of the AC signal increases, the rate of change of the magnetic field also increases. This leads to stronger induced eddy currents. These eddy currents more effectively counteract the current flow in the center of the conductor, forcing the main current to flow in a progressively thinner layer near the conductor's surface.

Therefore, the skin effect becomes significantly more pronounced as the frequency of the alternating current increases.

Skin Depth Explained

The extent of the skin effect is often quantified by a parameter called the skin depth, denoted by the Greek letter delta ($\delta$). Skin depth is defined as the depth below the conductor's surface where the current density drops to approximately 37% (or $1/e$) of its value at the surface.

The formula for skin depth is approximately:

$\delta = \sqrt{\frac{2\rho}{\omega\mu}} = \sqrt{\frac{\rho}{\pi f \mu}}$

Where:

• $\rho$ (rho) is the electrical resistivity of the conductor material.
• $\omega$ (omega) is the angular frequency of the AC ($ \omega = 2 \pi f $).
• $f$ is the frequency of the AC.
• $\mu$ (mu) is the magnetic permeability of the conductor material.

From the formula, we can see that the skin depth ($\delta$) is inversely proportional to the square root of the frequency ($f$):

$\delta \propto \frac{1}{\sqrt{f}}

This mathematical relationship clearly shows that as the frequency ($f$) increases, the skin depth ($\delta$) decreases. A smaller skin depth signifies that the current is confined to a shallower region near the surface, making the skin effect more prominent.

Key Takeaway

In summary, for an alternating current in an electrical conductor, the skin effect becomes more prominent (i.e., the current flows in a thinner layer near the surface) as the frequency of the AC signal increases.