Electromagnetism Statements Analysis

The question asks to identify the incorrect statements regarding electromagnetism concepts like intrinsic resistance, frequency effects, and skin depth.

Statement 1 Analysis: Intrinsic Resistance and Frequency

Statement 1 claims: "Intrinsic resistance is a function of frequency and it is less than that for a perfect dielectric." Let's interpret "intrinsic resistance" as the magnitude of the characteristic impedance ($\vert\eta\vert$) of a medium.

• The characteristic impedance of a lossy medium is given by $\eta = \sqrt{\frac{j\omega\mu}{\sigma + j\omega\epsilon}}$.
• Its magnitude is $\vert\eta\vert = \sqrt{\frac{\omega^2\mu^2}{\sigma^2 + \omega^2\epsilon^2}} = \sqrt{\frac{\mu}{\epsilon}} \sqrt{\frac{\omega^2\epsilon^2}{\sigma^2 + \omega^2\epsilon^2}} = \eta_{\text{perfect}} \sqrt{\frac{1}{1 + (\sigma/\omega\epsilon)^2}}$.
• This shows $\vert\eta\vert$ is indeed a function of frequency ($\omega$) and conductivity ($\sigma$).
• For a lossy medium ($\sigma > 0$), the term $\sqrt{\frac{1}{1 + (\sigma/\omega\epsilon)^2}}$ is less than 1. Thus, $\vert\eta\vert_{\text{lossy}} < \eta_{\text{perfect}}$.
• Therefore, statement 1 is correct.

Statement 2 Analysis: Frequency, Loss, and Intrinsic Resistance

Statement 2 claims: "An increase in the frequency or loss in the dielectric results in the lowering of intrinsic resistance."

• From the formula $\vert\eta\vert = \eta_{\text{perfect}} \sqrt{\frac{1}{1 + (\sigma/\omega\epsilon)^2}}$:
  • Effect of Frequency ($\omega$): As $\omega$ increases, the ratio $\sigma/(\omega\epsilon)$ decreases. This makes the term $\sqrt{\frac{1}{1 + (\sigma/\omega\epsilon)^2}}$ increase, causing $\vert\eta\vert$ to increase (approaching $\eta_{\text{perfect}}$). This contradicts the statement.
  • Effect of Loss ($\sigma$): As $\sigma$ increases (higher loss), the ratio $\sigma/(\omega\epsilon)$ increases. This makes the term $\sqrt{\frac{1}{1 + (\sigma/\omega\epsilon)^2}}$ decrease, causing $\vert\eta\vert$ to decrease. This part aligns with the statement.
• Since increasing frequency *increases* intrinsic resistance (magnitude), statement 2 is incorrect.

Statement 3 Analysis: Skin Depth Definition

Statement 3 claims: "In high-frequency case, the skin depth represents the depth at which the electric intensity is $0.707$ of its surface value."

• The standard definition of skin depth ($\delta$) is the depth at which the amplitude of the electric or magnetic field decays to $1/e$ (approximately $0.368$) of its surface value.
• The value $0.707$ (or $1/\sqrt{2}$) represents the point where the power drops to half, not the field amplitude.
• Therefore, statement 3 is incorrect.

Conclusion

Statements 2 and 3 are incorrect.