Electromagnetic Waves

This chapter introduces electromagnetic (EM) waves, which are time-varying electric and magnetic fields that propagate through space. It begins with the concept of displacement current, Maxwell's correction to Ampere's law, which shows that a changing electric field can produce a magnetic field. This insight unifies electricity and magnetism and predicts the existence of EM waves. You will study the properties of these waves: they are transverse, travel at the speed of light in a vacuum, and do not require a medium for propagation.

The core of the chapter involves the mathematical description of a plane EM wave, including the sinusoidal expressions for the electric field (E) and magnetic field (B). A key skill is understanding the relationship between the directions of E, B, and the wave's propagation. You will also learn to calculate the energy carried by these waves using concepts like energy density and the Poynting vector, which describes the flow of energy. The chapter concludes with the electromagnetic spectrum, which classifies EM waves like radio waves, microwaves, and X-rays based on their frequency.

This is a relatively short and theoretical chapter, but the concepts can be abstract. The main difficulty lies in visualizing the three-dimensional relationship between the electric field, magnetic field, and direction of travel, which often involves vector cross products. The chapter builds directly on your understanding of Gauss's law, Ampere's law, and Faraday's law from previous chapters on electricity and magnetism.

Formulas for Electromagnetic Waves

Displacement current: $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$ \Phi_E is the electric flux.
Ampere-Maxwell law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 (I_c + \epsilon_0 \frac{d\Phi_E}{dt})$ I_c is the conduction current through the loop.
Speed of an EM wave in vacuum: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}}$ \mu_0 is permeability and \epsilon_0 is permittivity of free space.
Speed of an EM wave in a medium: $v = \frac{1}{\sqrt{\mu \epsilon}}$ \mu is permeability and \epsilon is permittivity of the medium.
Relation between E and B field amplitudes: $E_0 = c B_0$ This relation holds for the amplitudes of the fields in vacuum.
Poynting vector: $\vec{S} = \frac{1}{\mu_0} (\vec{E} \times \vec{B})$ Gives the direction and rate of energy flow per unit area.
Average intensity of an EM wave: $I = \frac{1}{2} c \epsilon_0 E_0^2 = \frac{c B_0^2}{2 \mu_0}$ E_0 and B_0 are the maximum amplitudes of the fields.
Average total energy density: $u_{avg} = \frac{1}{2} \epsilon_0 E_0^2$ The average electric and magnetic energy densities are equal.
Momentum transferred (complete absorption): $p = \frac{U}{c}$ U is the total energy of the wave absorbed by the surface.
Radiation pressure (complete absorption): $P_{rad} = \frac{I}{c}$ I is the intensity of the incident wave. For perfect reflection, P_rad = 2I/c.