# The wave nature of light

Light is an electromagnetic wave composed of electric and magnetic components.

I recently read that the velocity of light is in the direction $$\mathbf{E}\times\mathbf{B}$$ where $$\mathbf{E}$$ and $$\mathbf{B}$$ are the electric and magnetic field vectors.

Now, suppose at some point in space, the electric and magnetic fields are $$\mathbf{E}$$ and $$\mathbf{B}$$, which are both functions of time, suppose that we place a mirror there, perpendicular to the incident light, then, due to reflection from denser medium, there will be a phase difference of $$\pi$$ in the electric and magnetic field components of the wave. So, the electric and magnetic fields are $$-\mathbf{E}, -\mathbf{B}$$ after reflection. Hence, the direction of propagation of light after reflection is the same as the direction of $$(\mathbf{-E})\times(\mathbf{-B})=\mathbf{E}\times\mathbf{B}$$ which means that the light wave is still traveling in the same direction.

Where is the error in this?

From Wikipedia, Reflection Phase Change

"Phase" here is the phase of the electric field oscillations, not the magnetic field oscillations (while the electric field will undergo 180° phase change, the magnetic field will undergo 0° phase change.

Suppose interface to be $$xy$$-plane then $$\mathbf{E}_I(z,t)=E_{0I} \ e^{i(k_1z-\omega t)}\hat{x}$$ $$\mathbf{B}_I(z,t)=\frac{1}{v_1}E_{0I} \ e^{i(k_1z-\omega t)}\hat{y}$$

It gives rise to a reflected wave, $$\mathbf{E}_R(z,t)=E_{0R} \ e^{i(-k_1z-\omega t)}\hat{x}$$ $$\mathbf{B}_R(z,t)=-\frac{1}{v_1}E_{0R} \ e^{i(-k_1z-\omega t)}\hat{y}$$ Now if $$v_2 then there will be phase difference introducing this to reflected waves $$\mathbf{E}_R(z,t)=E_{0R} \ e^{i(-k_1z-\omega t+\pi)}\hat{x}$$ $$\mathbf{B}_R(z,t)=-\frac{1}{v_1}E_{0R} \ e^{i(-k_1z-\omega t+\pi)}\hat{y}$$

$$\mathbf{E}_R(z,t)=-E_{0R} \ e^{i(-k_1z-\omega t)}\hat{x}$$ $$\mathbf{B}_R(z,t)=\frac{1}{v_1}E_{0R} \ e^{i(-k_1z-\omega t)}\hat{y}$$

The direction of wave propagation: $$-\hat{x}\times\hat{j}=-\hat{k}$$

It should be.

Reference : Sec. 9.3.2 Electrodynamics : DJ Griffith

• Also, can we explain it using the fact that magnetic field is a pseudo vector? Sep 28 at 9:29