JEE Advanced EM Waves

Cheatsheet Content

Electromagnetic Waves: Basics Definition: EM waves are transverse waves consisting of oscillating electric ($\vec{E}$) and magnetic ($\vec{B}$) fields, perpendicular to each other and to the direction of propagation. Source: Accelerated or oscillating charges produce EM waves. Nature: Transverse, non-mechanical (do not require a medium), travel at the speed of light in vacuum. Maxwell's Equations (Integral Form) These equations form the foundation of electromagnetism: Gauss's Law for Electricity: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$ Relates electric flux to enclosed charge. Gauss's Law for Magnetism: $\oint \vec{B} \cdot d\vec{A} = 0$ Magnetic monopoles do not exist; magnetic field lines are always closed loops. Faraday's Law of Induction: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$ A changing magnetic flux induces an electric field. Ampere-Maxwell Law: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ A changing electric flux (displacement current $I_D = \epsilon_0 \frac{d\Phi_E}{dt}$) or conduction current ($I_{enc}$) produces a magnetic field. Properties of EM Waves Speed in Vacuum: $c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$ Speed in Medium: $v = \frac{1}{\sqrt{\mu \epsilon}} = \frac{c}{\sqrt{\mu_r \epsilon_r}} = \frac{c}{n}$ Where $n$ is the refractive index of the medium. Relationship between E and B: In vacuum, $E_0 = c B_0$ (for peak values) or $E = cB$ (for instantaneous values). Direction of Propagation: $\vec{E} \times \vec{B}$ gives the direction of wave propagation. Energy Density: Electric field energy density: $u_E = \frac{1}{2} \epsilon_0 E^2$ Magnetic field energy density: $u_B = \frac{1}{2\mu_0} B^2$ Total energy density: $u = u_E + u_B = \epsilon_0 E^2 = \frac{1}{\mu_0} B^2$ (in vacuum) Poynting Vector: $\vec{S} = \frac{1}{\mu_0} (\vec{E} \times \vec{B})$ Represents the energy flux (power per unit area) transported by the EM wave. Magnitude: $S = \frac{EB}{\mu_0}$. Average intensity: $I_{avg} = \langle S \rangle = \frac{E_0 B_0}{2\mu_0} = \frac{E_0^2}{2\mu_0 c} = \frac{c B_0^2}{2\mu_0}$. Momentum: EM waves carry momentum. Momentum $p = \frac{U}{c}$ (for total energy $U$). Radiation pressure $P_R = \frac{I}{c}$ (for perfect absorption) or $P_R = \frac{2I}{c}$ (for perfect reflection). Wave Equation for EM Waves In vacuum, the electric and magnetic fields satisfy the wave equation: $\frac{\partial^2 E_x}{\partial z^2} = \mu_0 \epsilon_0 \frac{\partial^2 E_x}{\partial t^2}$ $\frac{\partial^2 B_y}{\partial z^2} = \mu_0 \epsilon_0 \frac{\partial^2 B_y}{\partial t^2}$ For a plane EM wave traveling in the +z direction: $\vec{E} = E_0 \sin(kz - \omega t) \hat{i}$ $\vec{B} = B_0 \sin(kz - \omega t) \hat{j}$ Where $k = \frac{2\pi}{\lambda}$ (wave number) and $\omega = 2\pi f$ (angular frequency). $v = \frac{\omega}{k} = \lambda f = c$ (in vacuum). Electromagnetic Spectrum EM waves are classified by their frequency/wavelength. All travel at speed $c$ in vacuum. Region Wavelength Range Frequency Range Applications/Properties Radio Waves $> 0.1 \text{ m}$ $ Radio, TV communication, MRI Microwaves $0.1 \text{ m} - 1 \text{ mm}$ $3 \times 10^9 - 3 \times 10^{11} \text{ Hz}$ Microwave ovens, radar, satellite communication Infrared (IR) $1 \text{ mm} - 700 \text{ nm}$ $3 \times 10^{11} - 4 \times 10^{14} \text{ Hz}$ Heaters, remote controls, night vision, optical fibers Visible Light $700 \text{ nm} - 400 \text{ nm}$ $4 \times 10^{14} - 7.5 \times 10^{14} \text{ Hz}$ Human vision, lasers, photography Ultraviolet (UV) $400 \text{ nm} - 1 \text{ nm}$ $7.5 \times 10^{14} - 3 \times 10^{17} \text{ Hz}$ Sterilization, tanning, water purification X-rays $1 \text{ nm} - 10^{-3} \text{ nm}$ $3 \times 10^{17} - 3 \times 10^{20} \text{ Hz}$ Medical imaging, security scanning, crystallography Gamma Rays $ $> 3 \times 10^{20} \text{ Hz}$ Nuclear reactions, cancer therapy, astronomy Key Concepts for JEE Advanced Displacement Current: Maxwell corrected Ampere's law by adding the displacement current term $\epsilon_0 \frac{d\Phi_E}{dt}$. It exists even in vacuum where no conduction charge flows, e.g., between capacitor plates during charging/discharging. Energy and Momentum Transfer: EM waves carry energy and momentum, which leads to radiation pressure. This is important in topics like solar sails. Polarization: The direction of the electric field vector determines the polarization of the EM wave. Unpolarized light: $\vec{E}$ oscillates in all possible directions perpendicular to propagation. Plane polarized light: $\vec{E}$ oscillates in a single plane. Intensity: Proportional to $E_0^2$ and $B_0^2$. $I \propto A^2$ (where $A$ is amplitude). Qualitative Understanding: Be able to draw the $\vec{E}$ and $\vec{B}$ field oscillations for a given direction of propagation. Remember they are in phase and perpendicular to each other and to the propagation direction.