Electromagnetic Waves

Derivation of Wave Equation for Electric Field in Vacuum

Conditions: Free space (charge density $\rho = 0$, current density J = 0)

• Start with Faraday's law in differential form: \nabla \times $E = -\frac{\partial B}{\partial t}$
• Take the curl of both sides: \nabla \times (\nabla \times E) = \nabla \times \left(-\frac{\partial B}{\partial t}\right) = -\frac{\partial}{\partial t}(\nabla \times B)
• Use the vector identity \nabla \times (\nabla \times E) = \nabla(\nabla \cdot E) - \nabla^{2}E and noting that ∇ · E = 0 in free space, obtaining -\nabla^{2}$E = -\frac{\partial}{\partial t}(\nabla \times B)$
• Substitute Ampere-Maxwell law in free space \nabla \times $B = \mu_{0}\varepsilon_{0}\frac{\partial E}{\partial t} into the equation$
• Resulting in \nabla^{2}$E = \mu_{0}\varepsilon_{0} \frac{\partial^{2}E}{\partial t^{2}}$

Result: \nabla^{2}$E = \mu_{0}\varepsilon_{0} \frac{\partial^{2}E}{\partial t^{2}} with wave speed v = \frac{1}{\sqrt{\mu_{0}\varepsilon_{0}}}$

Relation between Electric and Magnetic Field Amplitudes

Conditions: Monochromatic plane electromagnetic wave propagating in vacuum along z-axis

• Let the electric field be $E = E_{0} \sin(kz - \omega t)\hat{i}$ and magnetic field be $B = B_{0} \sin(kz - \omega t)\hat{j}$
• Apply Faraday's law \nabla \times $E = -\frac{\partial B}{\partial t} in Cartesian coordinates$
• Evaluate the curl of E: \frac{\partial E_{x}}{\partial z} \hat{j} = k E_{0} \cos(kz - \omega t)\hat{j}
• Evaluate the time derivative of B: -\frac{\partial B}{\partial t} = \omega B_{0} \cos(kz - \omega t)\hat{j}
• Equate the components: k $E_{0} = \omega B_{0}$, which simplifies using \frac{\omega}{k} = c

Result: $E_{0} = c B_{0} \text{ or } \frac{E}{B} = c$

Orthogonal Sinusoidal Oscillations of E and B Fields

Draw: A coordinate system with z-axis as propagation direction, showing an electric field oscillating along the y-axis (sine wave) and a magnetic field oscillating along the x-axis (cosine wave in phase with E).

Interpret: Demonstrates the transverse nature and spatial-temporal quadrature relationship of the coupled fields.

Exam use: Frequently asked in TGT/PGT to identify field orientations and polarization directions.

Electromagnetic Spectrum Chart

Draw: A logarithmic scale spanning from 10^-18 m to 10⁴ m, marking regions of gamma rays, X-rays, UV, visible, IR, microwaves, and radio waves with increasing wavelength.

Interpret: Shows the inverse relation between frequency and wavelength across the spectrum.

Exam use: Direct factual questions on ordering of frequencies, wavelengths, and detection methods.

Hertz's Experimental Setup

Draw: An induction coil connected to spark gap transmitter spheres, paired with a distant wire loop acting as a receiver spark gap.

Interpret: Proved the existence of electromagnetic waves by observing induced sparks across the receiver gap via resonance.

Exam use: Tested conceptually for historical context and verification of Maxwell's theory.

The amplitude of the magnetic field part of a harmonic electromagnetic wave in vacuum is $B_{0} = 510 nT$. What is the amplitude of the electric field part of the wave?

• Identify given values: $B_{0} = 510 \times 10^{-9} T$, $c = 3 \times 10^{8} m/s$.
• Use the relation between electric and magnetic field amplitudes: $E_{0} = c B_{0}$.
• Substitute values: $E_{0} = (3 \times 10^{8} m/s) \times (510 \times 10^{-9} T)$.
• Calculate result: $E_{0} = 153 V/m$.

Answer: 153 V/m

Light with an energy flux of 18 W/cm^{2} falls on a non-reflecting surface at normal incidence. If the surface has an area of 20 cm^{2}, find the average total momentum delivered to the surface during a 30-minute time span.

• Convert intensity to SI units: $I = 18 W/cm^{2} = 18 \times 10^{4} W/m^{2} = 1.8 \times 10^{5} W/m^{2}$.
• Calculate total energy incident: U = I \times A \times t where $A = 20 \times 10^{-4} m^{2}$ and t = 30 \times 60 = 1800 s.
• $U = (1.8 \times 10^{5}) \times (20 \times 10^{-4}) \times 1800 = 6.48 \times 10^{5} J$.
• For complete absorption, momentum $p = U / c = \frac{6.48 \times 10^{5}}{3 \times 10^{8}}$.
• Calculate final momentum: $p = 2.16 \times 10^{-3} kg \cdot m/s$.

Answer: 2.16 × 10^{-3} kg·m/s

In a plane electromagnetic wave, the electric field oscillates sinusoidally at a frequency of 2.0 × 10^{10} Hz and an amplitude of 48 V/m. Calculate the average energy density of the electromagnetic wave.

• Identify given values: $E_{0} = 48 V/m$, \varepsilo$n_{0} = 8.854 \times 10^{-12} C^{2}/N \cdot m^{2}$.
• Recall the formula for average energy density: $u_{avg} = \frac{1}{2} \varepsilon_{0} E_{0}^{2}$.
• Substitute values: $u_{avg} = \frac{1}{2} \times (8.854 \times 10^{-12}) \times (48)^{2}$.
• Calculate: $u_{avg} = 0.5 \times 8.854 \times 10^{-12} \times 2304 = 1.02 \times 10^{-8} J/m^{3}$.

Answer: 1.02 × 10^{-8} J/m^{3}