📝 Chapter Notes & Revision

Electromagnetic Waves

🏫 MP BoardClass 12Physics

📐 Formula & Cheat Sheet (English)

Class 12 Physics: Electromagnetic Waves (वैद्युतचुंबकीय तरंगें)

Quick Revision Notes & Formula Sheet (MP Board)

1. Displacement Current (विस्थापन धारा)

Displacement current arises due to a time-varying electric field (or changing electric flux). It was introduced by James Clerk Maxwell to remove the inconsistency in Ampere’s Circuital Law.

  • Formula: $$I_d = \epsilon_0 \frac{d\Phi_E}{dt}$$ Where:

    • $I_d$ = Displacement current
    • $\epsilon_0$ = Permittivity of free space ($8.854 \times 10^{-12} \text{ C}^2/\text{N}\cdot\text{m}^2$)
    • $\Phi_E$ = Electric flux through the surface = $\oint \vec{E} \cdot d\vec{A}$

  • Total Current: $$I_{total} = I_c + I_d = I_c + \epsilon_0 \frac{d\Phi_E}{dt}$$ (In a conduction path, $I_d = 0$; inside a capacitor during charging, $I_c = 0$ and $I_d = I_c$)

2. Maxwell's Equations (मैक्सवेल के समीकरण)

Maxwell summarized all basic laws of electricity and magnetism into four fundamental equations:

  1. Gauss's Law in Electrostatics: $$\oint \vec{E} \cdot d\vec{A} = \frac{q}{\epsilon_0}$$

  2. Gauss's Law in Magnetism: $$\oint \vec{B} \cdot d\vec{A} = 0 \quad \text{(Monopoles do not exist)}$$

  3. Faraday's Law of Electromagnetic Induction: $$\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$$

  4. Ampere-Maxwell Law: $$\oint \vec{B} \cdot d\vec{l} = \mu_0 \left( I_c + \epsilon_0 \frac{d\Phi_E}{dt} \right)$$

3. Properties & Formulas of Electromagnetic Waves

EM waves are transverse waves produced by accelerated charges. They consist of sinusoidally varying electric ($\vec{E}$) and magnetic ($\vec{B}$) fields oscillating perpendicular to each other and perpendicular to the direction of wave propagation.

Mathematical Representation:

If wave propagates along the +X direction:

  • Electric field: $E_y = E_0 \sin(kx - \omega t)$
  • Magnetic field: $B_z = B_0 \sin(kx - \omega t)$

Where:

  • Wave number: $k = \frac{2\pi}{\lambda}$
  • Angular frequency: $\omega = 2\pi\nu$
  • Wave speed: $v = \frac{\omega}{k} = \nu \lambda$

Speed of EM Waves:

  • In Vacuum / Free Space: $$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \approx 3 \times 10^8 \text{ m/s}$$ Where: $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$, $\epsilon_0 = 8.854 \times 10^{-12} \text{ F/m}$

  • In a Medium: $$v = \frac{1}{\sqrt{\mu \epsilon}} = \frac{c}{\sqrt{\mu_r \epsilon_r}} = \frac{c}{n}$$ Where: $n = \sqrt{\mu_r \epsilon_r}$ is the refractive index of the medium.

Relation between Electric and Magnetic Field Amplitudes:

$$c = \frac{E_0}{B_0} = \frac{E_{rms}}{B_{rms}}$$

4. Energy Density, Intensity, and Momentum

Energy Density ($u$):

Energy stored per unit volume in an EM wave:

  • Electric energy density: $u_E = \frac{1}{2} \epsilon_0 E^2$
  • Magnetic energy density: $u_B = \frac{B^2}{2\mu_0}$

Average Energy Density ($u_{avg}$): $$u_{avg} = u_E + u_B = \frac{1}{4} \epsilon_0 E_0^2 + \frac{B_0^2}{4\mu_0} = \frac{1}{2} \epsilon_0 E_0^2 = \frac{B_0^2}{2\mu_0}$$ (Since average electric energy density equals average magnetic energy density: $u_E = u_B$)

Intensity of EM Wave ($I$):

Radiant energy passing per unit area per second normal to the direction of propagation. $$I = u_{avg} \cdot c = \frac{1}{2} \epsilon_0 E_0^2 c = E_{rms}^2 \epsilon_0 c$$

Momentum ($p$) & Radiation Pressure ($P$):

When an EM wave of total energy $U$ strikes a surface:

  • Linear Momentum Transferred:

    • For Complete Absorption: $p = \frac{U}{c}$
    • For Complete Reflection: $p = \frac{2U}{c}$

  • Radiation Pressure:

    • For Complete Absorption: $P = \frac{I}{c}$
    • For Complete Reflection: $P = \frac{2I}{c}$

5. Electromagnetic Spectrum (वैद्युतचुंबकीय स्पेक्ट्रम)

Arranged in order of increasing frequency ($\nu$) / decreasing wavelength ($\lambda$):

Type of WaveWavelength Range ($\lambda$)Frequency Range ($\nu$)Source / ProductionPrimary Application / Uses
Radio Waves$> 0.1 \text{ m}$$< 3 \times 10^9 \text{ Hz}$Accelerated motion of charges in conducting wiresRadio & TV communication, Cellular phones
Microwaves$0.1 \text{ m}$ to $1 \text{ mm}$$10^9 \text{ Hz}$ to $10^{11} \text{ Hz}$Special vacuum tubes (Klystron, Magnetron)Radar systems, Microwave ovens, Satellite communication
Infrared (Heat waves)$1 \text{ mm}$ to $700 \text{ nm}$$10^{11} \text{ Hz}$ to $4 \times 10^{14} \text{ Hz}$Hot bodies and moleculesNight vision, Remote controls, Physiotherapy, Greenhouse effect
Visible Light$700 \text{ nm}$ to $400 \text{ nm}$$4 \times 10^{14} \text{ Hz}$ to $8 \times 10^{14} \text{ Hz}$Atomic excitation, Electrons moving in atomsProvides vision, Optical instruments
Ultraviolet (UV)$400 \text{ nm}$ to $1 \text{ nm}$$8 \times 10^{14} \text{ Hz}$ to $10^{16} \text{ Hz}$Very hot bodies, Sun, Arc lampsWater purification (germicidal), Checking forgery, LASIK eye surgery
X-Rays$1 \text{ nm}$ to $10^{-3} \text{ nm}$$10^{16} \text{ Hz}$ to $10^{19} \text{ Hz}$Sudden stopping of fast-moving electrons on heavy targetDiagnostic tool in medicine (bone fractures), Crystal structure analysis
Gamma Rays ($\gamma$)$< 10^{-3} \text{ nm}$$> 10^{19} \text{ Hz}$Radioactive decay of atomic nucleiCancer therapy (radiotherapy), Destroying cancer cells

6. Quick Revision Shortcuts for Board Exams

  1. Order of Wavelengths (Longest to Shortest): Radio > Micro > Infrared > Visible > UV > X-rays > Gamma (Memory Trick: Radio Men Invaded Very Unique Xmas Gardens)

  2. Transverse Nature: $\vec{E} \cdot \vec{B} = 0$, $\vec{E} \cdot \vec{v} = 0$, and $\vec{B} \cdot \vec{v} = 0$.

  3. Poynting Vector ($\vec{S}$): Represents energy flow rate per unit area. $$\vec{S} = \frac{1}{\mu_0} (\vec{E} \times \vec{B})$$

  4. Direction of propagation: Always given by the cross product $\vec{E} \times \vec{B}$.