Boundary conditions for an imperfect conductor

Question

Lets say we have a wave which is linearly polarized and is incident to the surface of a imperfect conductor, which we will say is the plane $z=0$. Suppose the incident wave $E_i$ is parallel to the surface.

We know there will be a wave $E_r$ which is reflected, and because the conductor is imperfect, there will also be a transmitted wave $E_t$ which weakly penetrates weakly the conductor (skin effect).

To calculate reflection/transmission coefficients we need the boundary conditions. We have:

$E_i(z=0)+E_r(z=0)=E_t(z=0)$ because the electric field is continuos across the boundary in the direction tangentiel to the surface.
Assuming no surface courant, we should also have continuity of the magnetic field across the boundary: $B_i(z=0)+B_r(z=0)=B_t(z=0)$.

However, according to slide 10 of https://www2.ph.ed.ac.uk/~playfer/EMlect15.pdf, the second relation should actually be $B_i(z=0)-B_r(z=0)=B_t(z=0)$ (*).

The above link gives the correct answer for the relfection/transmission coefficients, so the equation (*) they give is not a typo.

I can't figure out for he life of me the source of the minus sign!

freecharly · Accepted Answer

This is easy to explain. The boundary condition in the linked lecture slide (p. 10) for the magnetic field with the minus sign before the reflected amplitude is wrong! Your boundary conditions 1. and 2. for the electric and magnetic field amplitudes are correct.

In the considered case, there are no surface charges or surface currents and the permeabilities are assumed to be $mu_0$ so that the necessary boundary conditions for wave reflection and transmission are the continuity of the tangential electric fields $$E_1=E_2 tag 1$$ and the continuity of the tangential magnetic fields $$B_1=B_2 tag 2$$ to which your correct amplitude boundary conditions 1. and 2 correspond.  With the (complex) vectorial electric field amplitude $vec E_0$, the incident electric wave is described by $$vec E=vec E_{0i} exp i( vec k vec r-omega t) tag 3$$ where $vec k=k_i hat z$ is the wave vector pointing in positive z-direction, $vec r = z hat z$ is the position vector, $omega$ is the angular frequency, and $t$ is the time. The reflected wave is $$vec E=vec E_{0r} exp i(- vec k vec r-omega t) tag 4$$ Maxwell's equation $$nabla times vec E=-frac {partial B}{partial t} tag 5$$ relates the vectorial magnetic field amplitude $vec B_0$ to the electrical amplitude $vec E_0$ by the cross product $$vec k times vec E_0=omegavec B_0 tag 6$$ Thus, when the electric field $vec E_{0i}$ of the incident wave is in positive x-direction, the magnetic field $vec B_{0i}$ is in positive y-direction giving $$B_{0i}= frac {k_i}{omega} E_{0i}tag 7$$ For the transmitted wave one obtains $$B_{0t}= frac {k_t}{omega} E_{0t} tag 8$$ The reflected wave propagates in negative z-direction so that the wave vector in eqs. (6) is $-vec k=-k_i hat z$ and the relation of the reflected magnetic and electric field amplitude has a negative sign $$B_{0r}= -frac {k_i}{omega} E_{0r} tag 9$$ Thus the boundary condition for the magnetic field (2) can be expressed by the electric field amplitudes $$E_{0i}-E_{0r}=frac {k_t}{k_{i}} E_{0t} tag {10} $$ Note the negative sign before $E_{0r}$! The boundary condition for the electric field (1) gives the other equation for the electric field amplitudes $$E_{0i}+E_{0r}=E_{0t} tag {11}$$ From eqs. (10) and (11) follow the (amplitude) reflection and transmission coefficients for any linear (not magnetic) materials 1 and 2. In particular, also for an imperfect conductor where $k_t$ is, in general, a complex quantity!

ProfRob · Answer

The boundary conditions in the lecture slides you refer to are perfectly correct.
Your condition (1) implies a situation where you consider (algebraically) that the incident, reflected and transmitted E-fields are all pointing in the same direction. The boundary condition that the E-fields tangential to the boundary must be continuous then gives you
$$E_i(z=0) + E_r(z=0) = E_t(z=0) .$$
However, the reflected wave is travelling in the opposite direction to the incident wave and $vec{E} times vec{B}$ gives the direction of wave travel.
This means that if the electric fields of the incident and refelected waves are initially defined to be in the same direction, then the directions of the magnetic fields of the incident and reflected waves must be opposite in order to be travelling in the appropriate direction.
The second relationship then results from applying the boundary condition that the tangential component of the H-field must be continuous across the interface,
$$ H_i(z=0) - H_r(z=0) = H_t(z=0) ,$$
that there are no surface currents and that the permeability is the same on both sides of the interface. The above equation inserts a minus sign in front of the reflected magnetic field because it is in the opposite direction to the incident magnetic field if $E_r$ is defined to be in the same direction as $E_i$.
Of course, when you do the maths, what you find is that if going from a high impedance medium (e.g. vacuum) to a low impedance medium (e.g. a good conductor), that the reflection coefficient is negative, meaning that the phase of the reflected wave is in fact reversed and the reflected E-field actually points in the opposite direction to the incident E-field, which is shown at the bottom of slide 10 in the linked presentation.

cms · Answer

I admit I didn't read the link you provided, but I am going to take a wild stab:

Reversing the sign on $vec B$ but not $vec E$ reverses the sign on the poynting vector $vec S$, thereby changing the direction of travel of the wave for the reflected wave.

Boundary conditions for an imperfect conductor

3 Answers

Add your own answers!

Ask a Question