Understanding Skin Depth and AC Frequency

The question explores the relationship between the frequency of an alternating current (AC) and the skin depth ($\delta$) in a conductor. The skin effect refers to the tendency of an AC electric current to flow predominantly near the surface (or "skin") of a conductor, rather than uniformly throughout its entire cross-section.

Formula for Skin Depth

The skin depth ($\delta$) is defined as the depth at which the current density decreases to $1/e$ (approximately 37%) of its value at the surface. It is determined by the properties of the conductor and the frequency of the AC signal. The formula for skin depth is:

$$ \delta = \sqrt{\frac{2}{\omega \mu \sigma}} $$

Where:

• $\delta$ represents the skin depth (in meters).
• $\omega$ is the angular frequency of the alternating current, related to the frequency ($f$) by $\omega = 2\pi f$ (in radians per second).
• $\mu$ is the magnetic permeability of the conductor material (in Henries per meter).
• $\sigma$ is the electrical conductivity of the conductor material (in Siemens per meter).

Relationship Between Frequency and Skin Depth

To understand how frequency affects skin depth, let's rewrite the formula in terms of frequency ($f$):

$$ \delta = \sqrt{\frac{2}{(2\pi f) \mu \sigma}} = \sqrt{\frac{1}{\pi f \mu \sigma}} $$

From this equation, we can see the relationship between skin depth ($\delta$) and frequency ($f$):

• $\delta$ is inversely proportional to the square root of the frequency ($f$).
• Mathematically, this is expressed as: $$ \delta \propto \frac{1}{\sqrt{f}} $$

This inverse relationship means that as the frequency ($f$) of the alternating current increases, the skin depth ($\delta$) must decrease. Conversely, as the frequency decreases, the skin depth increases, and the current penetrates deeper into the conductor.

Analyzing the Options

Based on the derived relationship:

• increases: This is incorrect because skin depth decreases as frequency increases.
• remains constant: This is incorrect as skin depth is dependent on frequency.
• decreases: This aligns with the formula ($\delta \propto 1/\sqrt{f}$), indicating that higher frequencies lead to shallower current penetration.
• becomes infinite: This would only happen as frequency approaches zero, not as it increases.

Conclusion

Therefore, as the frequency of an alternating current increases, the skin depth ($\delta$) in a conductor typically decreases.