Boundary Conditions at Interfaces

The Physical Intuition

Imagine a rope stretched horizontally. If you hold one end and create a wave, and the rope transitions into a heavier section, the wave does not vanish at the junction. Instead, part of the energy reflects back, and part transmits into the heavier section. The key physical constraint is that the rope itself cannot tear or overlap at the junction. Therefore, the vertical displacement of the rope must be continuous across the interface. In electromagnetism, the electric field plays a similar role to the displacement of the rope. If the tangential component of the electric field were discontinuous across a surface, it would imply an infinite electric field gradient, which is physically non-sensical for standard materials. Similarly, the magnetic field lines must form continuous loops; they cannot just start or stop at a boundary in the absence of magnetic monopoles. These intuitive requirements of continuity are what we formalise into the boundary conditions.

Deriving the Conditions from Integral Laws

To derive the boundary conditions, we apply the integral forms of Maxwell’s equations to a small "pillbox" or "loop" that straddles the interface. Consider a surface charge density \(\sigma_f\) and a surface current density \(\mathbf{K}_f\) at the interface. We define the normal vector \(\hat{\mathbf{n}}\) pointing from medium 1 to medium 2.

First, consider the normal component of the electric field. We apply Gauss’s law to a flat cylinder (pillbox) with its axis normal to the surface. As the height of the cylinder approaches zero, the flux through the curved surface vanishes. The flux through the top and bottom faces gives:

Thus, the normal component of the electric displacement field is discontinuous by the amount of the free surface charge density:

Next, consider the tangential component of the electric field. We apply Faraday’s law to a small rectangular loop lying in the plane of the interface. As the area of the loop approaches zero, the magnetic flux through the loop vanishes. The line integral of the electric field around the loop yields:

Therefore, the tangential component of the electric field must be continuous:

Magnetic Field Boundary Conditions

The conditions for the magnetic field are derived similarly. For the normal component, we use the fact that \(\nabla \cdot \mathbf{B} = 0\). Applying this to the same pillbox geometry, the net flux of \(\mathbf{B}\) through the closed surface must be zero. Since there are no magnetic monopoles, the flux entering one face must equal the flux leaving the other:

This means the normal component of the magnetic flux density is continuous across the interface.

For the tangential component, we apply Ampère’s law (with the displacement current term, which vanishes in the limit of zero area) to a small rectangular loop. The line integral of \(\mathbf{H}\) around the loop equals the conduction current passing through the loop. If there is a surface current density \(\mathbf{K}_f\) flowing along the interface, we get:

Vectorially, this is expressed as:

In most optical applications involving dielectric interfaces, there are no free surface currents (\(\mathbf{K}_f = \mathbf{0}\)), so the tangential component of \(\mathbf{H}\) is also continuous.

Application to Plane Waves

When a plane wave is incident on an interface, the total field in each medium is the sum of the incident and reflected (or transmitted) waves. The boundary conditions must hold for the total fields at every point on the interface and for all time. This imposes two critical constraints on the wave vectors and frequencies.

First, the time dependence must match. If the incident wave oscillates as \(e^{-i\omega t}\), the reflected and transmitted waves must have the same frequency \(\omega\); otherwise, the boundary conditions could not be satisfied for all \(t\).

Second, the spatial variation along the interface must match. The phase of the incident, reflected, and transmitted waves must be equal at every point on the boundary. This requires that the tangential components of the wave vectors be identical:

Since the magnitude of the wave vector is \(k = \frac{\omega}{v} = \frac{n\omega}{c}\), this condition leads directly to Snell’s law. Let \(\theta_i\), \(\theta_r\), and \(\theta_t\) be the angles of incidence, reflection, and transmission measured from the normal. The tangential component of \(\mathbf{k}\) is \(k \sin\theta\). Thus:

Similarly, comparing the incident and reflected waves in medium 1 gives \(n_1 \sin\theta_i = n_1 \sin\theta_r\), which implies \(\theta_r = \theta_i\), the law of reflection.