The depth of penetration of a wave in a lossy dielectric increases with increasingSelect one:a. Wavelengthb. Permittivityc. Permeabilityd. Conductivity

To determine which option affects the depth of penetration of a wave in a lossy dielectric, we need to analyze how each factor relates to wave propagation in such materials.

1. Break Down the Problem

• Understand the concept of depth of penetration in a lossy dielectric.
• Analyze each option: Wavelength, Permittivity, Permeability, Conductivity.

2. Relevant Concepts

• The depth of penetration (or skin depth) $\delta$ in a lossy dielectric is given by the formula: $\delta = \frac{1}{\sqrt{\pi f \mu \sigma}}$ Where:
• $f$ = frequency of the wave
• $\mu$ = permeability of the medium
• $\sigma$ = conductivity of the medium

3. Analysis and Detail

1. Wavelength: The relationship between wavelength and frequency is determined through the speed of light in the material. An increase in wavelength typically corresponds to a decrease in frequency, which can indirectly affect the depth of penetration through its relation to the $\sigma$ term in the equation.

2. Permittivity: Increasing permittivity can influence the phase velocity of the wave but does not directly correlate to an increase in penetration depth through the formula.

3. Permeability: Similar to permittivity, changing permeability can alter the wave's behavior but does not necessarily increase penetration depth.

4. Conductivity: Higher conductivity ($\sigma$) increases loss in the medium and, from the formula, leads to a decrease in depth of penetration ($\delta$). Thus, increasing conductivity decreases penetration depth.

4. Verify and Summarize

Analyzing the options, increasing permeability and permittivity may not contribute positively to penetration depth. However, an increase in wavelength translates to a decrease in frequency, which effectively increases the penetration depth given the relationship.

Final Answer

The depth of penetration of a wave in a lossy dielectric increases with increasing Wavelength (a).