# Boundary conditions for an imperfect conductor

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:

1. $E_i(z=0)+E_r(z=0)=E_t(z=0)$ because the electric field is continuos across the boundary in the direction tangentiel to the surface.

2. 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!

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 $$\vec k \times \vec E_0=\omega\vec B_0 \tag 6$$ 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!

• I believe this answer is false. First, the magnetic field condition does not, I believe, follow from the electric field condition, as the magnetic field condition depends on the surface courant! Second, the magnetic field of the reflected wave points in the same direction as the magnetic field of the incident wave at the boundary. Here's why... Apr 1, 2018 at 15:46
• Say the conductor is the region $z>0$. So the incident wave is $\vec E_i=E_i \exp(i (kz-\omega t)) \vec u_x$. The reflected wave is thus $\vec E_r=E_r exp(-i (kz+\omega t)) \vec u_x$. The incident magnetic field is $\vec B_i=(1/c)E_i \exp(i (kz-\omega t))\vec u_y$ and the reflected magnetic field is $\vec B_r=(1/c)E_r \exp(-i (kz+\omega t)) \vec u_y$. Apr 1, 2018 at 15:47
• @JoshuaBenabou - In the considered case of an imperfect conductor (i.e., with a conductivity $\sigma \neq \infty$), you have no surface current and thus , assuming $\mu_1=\mu_2$, the continuity boundary condition for the parallel magnetic fields $B1=B_2$ holds. This boundary condition doesn't follow from the continuity condition of the electric field $E_1=E_2$ equation. Thus the boundary conditions 1. and 2. stated in your question are correct! Apr 1, 2018 at 19:52
• @JoshuaBenabou - I looked at the lecture slides given in your link and it is obvious that the magnetic field boundary condition on page (10) with the minus sign $B_{0I}-B_{R}=B_{0T}$ is wrong, it should be a plus sign! However, the following relation for the electric field with the minus sign is correct, because again a sign error is made (in deriving $E_{0R}$ from $B_{0R}$ in not changing the sign of $k_1$ for the reflected wave) to obtain the correct result. Apr 1, 2018 at 19:56
• @JoshuaBenabou - I am sorry for hastily concluded correctness of the wrong magnetic field condition with the minus sign. I remembered the minus sign in the equation for the electric field amplitudes derived from the magnetic field continuity, which follows, as I correctly stated from the sign change of the wave vector of the reflected wave. I will edit my answer accordingly. Apr 1, 2018 at 20:04

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.

• I know it has to do something with the direction of the reflected wave, but I can't see exactly the problem. Apr 1, 2018 at 15:26

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.