Tutorial: Fields near the surface of a conductor

  1. (0 pts) Study the video lecture on the fields near the surface of a conductor and section 8.1 from Jackson. Recognize the importance of the small parameter . Remember that we are working with complex amplitudes of monochromatic fields, currents, and charge densities, as in \( j(r,t)=\mathop{\mathrm{Re}} j(r) \exp⁡(-i\omega t) \)
  2. (0 pts) Understand the meaning of Figs. 8.1 and 8.2 in Jackson.
  3. (3 pts) Reproduce the arguments showing that \( \frac{E_\parallel}{H_\parallel} \sim k\delta \).
  4. (3 pts) Reproduce the arguments showing that \( \frac{H_\perp}{H_\parallel} \sim k\delta \).
  5. (4 pts) Show that the normal component of the electric field is screened to first order in \( k \delta \); explicitly, this means that \( \frac{E_\perp^{in}}{E_\perp^{out}}\sim(k\delta)^2 \). To do this, use the first line of Eq. (8.5) and keep track of the orders of \( k\delta \): zeroth in \( \mathbf{H}_\parallel \), first in \( H_\perp \), and the fact that \( \nabla_t \sim k \) while \( \nabla_n\sim\delta^{-1} \). Also explain why \( E_\perp^{out}\sim c B_\parallel^{out} \).
  6. (0 pts) Observe that Fig. 8.2 shows \( E_\perp \) dropping abruptly to zero at the surface of the metal. This “zero” is, in fact, a small value of order \( (k\delta)^2 \), as follows from item #5.
  7. (2 pts) The discontinuity of \( E_\perp \) implies the existence of the surface charge density \( \Sigma \) strictly at the surface. Find \( \Sigma \) to first order in \( k\delta \), i.e., neglecting the small \( E_\perp \) inside the metal, from the Maxwell equation for \( \nabla\mathbf{E} \).
  8. (5 pts) Find \( \Sigma \) in a different way: (i) starting from Ohm’s law \( \mathbf{j}=\sigma\mathbf{E} \), find \( \nabla\mathbf{j} \) using the Maxwell equation for \( \nabla\mathbf{E} \) and taking into account that the conductivity is discontinuous at the surface; (ii) insert \( \nabla\mathbf{j} \) in the continuity equation and find the volume charge density from the resulting equation. You should find the charge density located strictly at the surface and vanishing identically inside the metal. Compare with the result of step #7 and identify the second-order correction in \( k \delta \).
  9. (2 pts) Using Eq. (8.11) for the leading term in \( \mathbf{E}_\parallel \) (or the fact that \( \mathbf{E}_\parallel \) is continuous at the surface), argue that the current density does not have a purely surface term (i.e., there is no “sheet current”).
  10. (4 pts) Show that the “surface current density” \( \mathbf{K} \) in Eq. (8.2) can be obtained by integrating the volume current density over the thickness of the metal. [Use Eq. 8.11 and Ohm’s law for this purpose, making sure that Eq. (8.2) is satisfied.]
  11. (0 pts) Ponder the fact that the surface charge density \( \Sigma \) in Eq. (8.1) is strictly at the surface of the metal (see steps #6-8), while the “surface current density” \( \mathbf{K} \) is spread over the skin depth.
  12. (2 pts) Because the surface charge density \( \Sigma \) oscillates in time while there is no surface current density, it follows that the current density must have a component \( j_\perp \) normal to the surface. Find \( j_\perp \) from the continuity equation.
  13. (2 pts) Show that \( \frac{j_\perp}{j_\parallel}\sim k\delta \) (your work on step #10 should be useful).