Table of Contents
How to Use This Page
Read each concept carefully, then check the formula, common mistake, and exam tip before moving to the next. This page covers Electromagnetic Waves completely for CBSE Class 12 Physics.
Key Concepts
Electromagnetic Waves
Core concepts you must know
1
Maxwell’s Equations
Formula
The four fundamental equations governing electromagnetism: Gauss’s Law for Electricity, Gauss’s Law for Magnetism, Faraday’s Law, and Ampere-Maxwell Law.
$$ \oint \vec{E} \cdot d\vec{s} = \frac{q}{\epsilon_0} \quad \text{(Gauss in Electrostatics)} $$
$$ \oint \vec{B} \cdot d\vec{s} = 0 \quad \text{(Gauss in Magnetism)} $$
$$ \oint \vec{E} \cdot d\vec{l} = -\frac{d}{dt} \int \vec{B} \cdot d\vec{s} \quad \text{(Faraday’s Law)} $$
$$ \oint \vec{B} \cdot d\vec{l} = \mu_0 I_c + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt} \quad \text{(Ampere-Maxwell Law)} $$
2
Velocity of EM Waves
Formula
EM waves travel through a vacuum at the speed of light ($c$). It is a universal constant determined by the permittivity ($\epsilon_0$) and permeability ($\mu_0$) of free space.
$$c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} = 3 \times 10^8 \text{ m/s}$$
3
Transverse Nature & Amplitudes
Formula
Electric and Magnetic fields oscillate sinusoidally, perpendicular to each other and to the direction of propagation. The direction of propagation is given by $\vec{E} \times \vec{B}$. The ratio of their amplitudes equals the speed of light.
$$E_0 = c B_0$$
4
Energy Density
Formula
EM waves carry energy. The energy is shared equally between the electric and magnetic fields. The total average energy density is the sum of both.
$$U_E = \frac{1}{4} \epsilon_0 E_0^2 \quad | \quad U_B = \frac{B_0^2}{4 \mu_0}$$
$$U_{\text{total}} = U_E + U_B = \frac{1}{2} \epsilon_0 E_0^2 = \frac{B_0^2}{2 \mu_0}$$
5
Momentum of EM Wave
Formula
EM waves carry momentum and can exert radiation pressure when they strike a surface. The linear momentum ($p$) is related to the energy ($U$) transferred.
$$p = \frac{U}{c} = \frac{h\nu}{c} = \frac{h}{\lambda}$$
Concept Deep Dive
01
The Missing Link: Displacement Current
How Maxwell fixed Ampere’s LawBefore Maxwell, Ampere’s law stated that only conduction current ($I_c$, actual moving electrons) creates a magnetic field. But consider a charging capacitor: between its plates, there are no moving electrons, yet a magnetic field exists!
Maxwell realized that a changing electric field (or changing electric flux) acts exactly like a current to produce a magnetic field. He called this the Displacement Current ($I_d$). This brilliant addition proved that changing electric and magnetic fields can sustain each other, allowing EM waves to travel through empty space.
Maxwell realized that a changing electric field (or changing electric flux) acts exactly like a current to produce a magnetic field. He called this the Displacement Current ($I_d$). This brilliant addition proved that changing electric and magnetic fields can sustain each other, allowing EM waves to travel through empty space.
$$I_d = \epsilon_0 \frac{d\Phi_E}{dt}$$
02
The Electromagnetic Spectrum
Arranging light by its wavelengthThe EM spectrum is the continuous distribution of electromagnetic waves arranged in order of wavelength or frequency. They all travel at the same speed ($c$) in a vacuum.
In order of Increasing Wavelength ($\lambda$) / Decreasing Frequency ($\nu$):
1. Gamma Rays: ($< 10^{-11} \text{ m}$) - Emitted by radioactive nuclei.
2. X-Rays: ($10^{-11}$ to $10^{-8} \text{ m}$) – Produced by stopping fast-moving electrons on high atomic number targets.
3. Ultraviolet (UV): ($10^{-8}$ to $4 \times 10^{-7} \text{ m}$) – Comes from the sun; causes tanning.
4. Visible Light: ($4 \times 10^{-7}$ to $7 \times 10^{-7} \text{ m}$) – The tiny band our eyes can detect.
5. Infrared (IR): ($7 \times 10^{-7}$ to $10^{-3} \text{ m}$) – Heat radiation emitted by warm bodies.
6. Microwaves: ($10^{-3}$ to $0.1 \text{ m}$) – Used in radar systems and cooking.
7. Radio Waves: ($> 0.1 \text{ m}$) – Produced by LC oscillators; used for broadcasting and communication.
In order of Increasing Wavelength ($\lambda$) / Decreasing Frequency ($\nu$):
1. Gamma Rays: ($< 10^{-11} \text{ m}$) - Emitted by radioactive nuclei.
2. X-Rays: ($10^{-11}$ to $10^{-8} \text{ m}$) – Produced by stopping fast-moving electrons on high atomic number targets.
3. Ultraviolet (UV): ($10^{-8}$ to $4 \times 10^{-7} \text{ m}$) – Comes from the sun; causes tanning.
4. Visible Light: ($4 \times 10^{-7}$ to $7 \times 10^{-7} \text{ m}$) – The tiny band our eyes can detect.
5. Infrared (IR): ($7 \times 10^{-7}$ to $10^{-3} \text{ m}$) – Heat radiation emitted by warm bodies.
6. Microwaves: ($10^{-3}$ to $0.1 \text{ m}$) – Used in radar systems and cooking.
7. Radio Waves: ($> 0.1 \text{ m}$) – Produced by LC oscillators; used for broadcasting and communication.
Compare & Contrast
✗ Mechanical Waves (e.g., Sound)
- Require a material medium to travel (cannot travel in vacuum).
- Can be transverse (string) or longitudinal (sound).
- Speed depends heavily on the medium’s elasticity and density.
- Speed in air is relatively slow ($\approx 343 \text{ m/s}$).
✓ Electromagnetic Waves (e.g., Light)
- Do NOT require any material medium (can travel perfectly in vacuum).
- Are strictly transverse in nature.
- Speed in vacuum is an absolute constant ($c$).
- Speed is incredibly fast ($3 \times 10^8 \text{ m/s}$).
Remember
When an EM wave enters a different medium (like from air to glass), its speed ($v$) and wavelength ($\lambda$) decrease, but its frequency ($\nu$) remains absolutely constant.
Common Mistakes to Avoid
Mistake 1
Mixing up the Vector Cross Product Order: The direction of propagation is explicitly $\vec{E} \times \vec{B}$. It is NEVER $\vec{B} \times \vec{E}$. If $\vec{E}$ is along the y-axis ($\hat{j}$) and $\vec{B}$ is along the z-axis ($\hat{k}$), the wave travels along $\hat{j} \times \hat{k} = \hat{i}$ (the x-axis).
Mistake 2
Using Peak Values instead of RMS for Average Energy: The formulas $U_E = \frac{1}{2} \epsilon_0 E^2$ use the RMS values of the field. If you are given the Peak amplitude ($E_0$), the formula becomes $U_E = \frac{1}{4} \epsilon_0 E_0^2$. Be very careful with the $1/2$ vs $1/4$ fractions!
Mistake 3
Misinterpreting Gauss’s Law in Magnetism: $\oint \vec{B} \cdot d\vec{s} = 0$ does NOT mean the magnetic field is zero. It means the net magnetic flux through any closed surface is zero. Whatever magnetic field lines enter the surface must exactly exit it, proving magnetic monopoles don’t exist.
Exam Tips
Tip 1
Mnemonic for the EM Spectrum: To remember the order from lowest energy (longest $\lambda$) to highest energy (shortest $\lambda$), use: “Red Martians Invade Venus Using X-ray Guns” (Radio, Micro, IR, Visible, UV, X-ray, Gamma).
Tip 2
In questions asking you to match a wavelength to its practical use:
• Radar/Aircraft navigation $\rightarrow$ Microwaves
• Treating muscular pain / TV remotes $\rightarrow$ Infrared
• Lasik eye surgery / Water purification $\rightarrow$ Ultraviolet
• Cancer treatment $\rightarrow$ Gamma rays
• Radar/Aircraft navigation $\rightarrow$ Microwaves
• Treating muscular pain / TV remotes $\rightarrow$ Infrared
• Lasik eye surgery / Water purification $\rightarrow$ Ultraviolet
• Cancer treatment $\rightarrow$ Gamma rays
Expected Exam Questions
SQ
Board Pattern Questions
Class 12 · EM Waves · CBSE Exam1
Which part of the electromagnetic spectrum has the largest penetrating power? Write its one use. [1 mark]
Answer
Gamma Rays
📝
Explanation
Gamma ($\gamma$) rays have the highest frequency and shortest wavelength, giving them the highest energy ($E = h\nu$) and largest penetrating power. They are commonly used in medicine for radiotherapy (destroying cancer cells).
2
The magnetic field in a plane electromagnetic wave is given by $B_y = 2 \times 10^{-7} \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t) \text{ T}$. What is the wavelength and frequency of the wave? [2 marks]
Answer
$\lambda = 1.25 \times 10^{-2} \text{ m}$, $\nu = 2.39 \times 10^{10} \text{ Hz}$
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Explanation
Compare the given equation with the standard wave equation: $B_y = B_0 \sin(kx + \omega t)$.
We get $k = 0.5 \times 10^3 \text{ rad/m}$ and $\omega = 1.5 \times 10^{11} \text{ rad/s}$.
Wavelength $\lambda = \frac{2\pi}{k} = \frac{2\pi}{0.5 \times 10^3} = 4\pi \times 10^{-3} \approx 1.25 \times 10^{-2} \text{ m}$.
Frequency $\nu = \frac{\omega}{2\pi} = \frac{1.5 \times 10^{11}}{2\pi} \approx 2.39 \times 10^{10} \text{ Hz}$.
3
A plane electromagnetic wave of frequency $25 \text{ MHz}$ travels in free space along the $x$-direction. At a particular point in space and time, $\vec{E} = 6.3 \hat{j} \text{ V/m}$. What is $\vec{B}$ at this point? [3 marks]
Answer
$\vec{B} = 2.1 \times 10^{-8} \hat{k} \text{ T}$
📝
Explanation
Magnitude: Using the relation $E = cB$, we have $B = \frac{E}{c}$.
$B = \frac{6.3}{3 \times 10^8} = 2.1 \times 10^{-8} \text{ T}$.
Direction: The wave propagates along the $x$-axis ($\hat{i}$), and the electric field is along the $y$-axis ($\hat{j}$). We know that direction of propagation is given by $\vec{E} \times \vec{B}$.
So, $\hat{i} = \hat{j} \times \text{direction of } \vec{B}$.
From vector cross products, $\hat{j} \times \hat{k} = \hat{i}$. Therefore, the magnetic field must be along the $z$-axis ($\hat{k}$).
Final Answer: $\vec{B} = 2.1 \times 10^{-8} \hat{k} \text{ T}$.
Concept Map
Electromagnetic Waves connects to →
Optics
Maxwell’s Equations
Displacement Current
Transverse Wave Nature
Radiation Pressure
Electromagnetic Spectrum
Photons
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