Electromagnetic Waves JEE Notes PDF: Download Now

Electromagnetic Waves JEE Notes: Important Concepts

Electromagnetic (EM) wave: a self-propagating combination of time-varying electric field $$\mathbf E$$ and magnetic field $$\mathbf B$$ produced whenever a charge accelerates. Both fields are:

• Mutually perpendicular and also perpendicular to the direction of wave travel (transverse nature).
• In perfect phase: the maxima and minima occur at the same instant and place.
• Linked by $$\dfrac{E_0}{B_0}=c$$ in free space, where $$c=3.00\times10^{8}\ \text{m s}^{-1}$$ is constant for all frequencies.

Wave parameters: frequency $$f$$ (or angular frequency $$\omega=2\pi f$$), wavelength $$\lambda$$, wave number $$k=2\pi/\lambda$$, and speed $$v$$. In a non-conducting medium of refractive index $$n$$, $$v=\dfrac{c}{n}$$.

How EM waves arise

1. An alternating current in an antenna makes electrons oscillate.
2. Oscillating charge ⇒ time-varying electric field.
3. Faraday-Lenz law: time-varying electric field induces magnetic field.
4. Ampere-Maxwell law: time-varying magnetic field in turn regenerates electric field.
5. The coupled fields detach from the source and travel outward as a wave.

Free-space wave equation

Starting from Maxwell equations (shown in the next section) one obtains identical wave equations:

$$\nabla^2 \mathbf E=\dfrac{1}{c^2}\dfrac{\partial^2 \mathbf E}{\partial t^2},\qquad \nabla^2 \mathbf B=\dfrac{1}{c^2}\dfrac{\partial^2 \mathbf B}{\partial t^2}.$$

General solution for a plane wave propagating along the +x axis:

$$\mathbf E=\hat y E_0\sin(kx-\omega t),\qquad \mathbf B=\hat z B_0\sin(kx-\omega t).$$

Relationship between field vectors

• Magnitude relation: $$E_0 = cB_0$$.
• Direction relation: $$\mathbf E\times \mathbf B$$ gives the direction of propagation.

Polarisation

If $$\mathbf E$$ vibrates in one fixed plane the wave is linearly polarised. EM waves from antennas are usually polarised; thermal radiation is generally unpolarised.

Maxwell’s Equations and Displacement Current

Integral form in free space (no free charge $$\rho=0$$, no free current $$\mathbf J=0$$)

Displacement current example

Parallel-plate capacitor, plate area $$A=40 \text{ cm}^2$$. Potential difference $$V=V_0\sin\omega t$$, with $$V_0=100\ \text{V},\; \omega=10^4\ \text{s}^{-1}$$ and separation $$d=2\ \text{mm}$$.

Find peak displacement current between the plates.

Capacitance $$C=\dfrac{\varepsilon_0 A}{d}= \dfrac{8.85\times10^{-12}\times4.0\times10^{-3}}{2\times10^{-3}}=1.77\times10^{-11}\ \text{F}$$.

Charge $$q=CV$$ ⇒ $$I_D=\dfrac{dq}{dt}=C\dfrac{dV}{dt}=C V_0\omega\cos\omega t$$.

Peak value $$I_{D0}=C V_0 \omega =1.77\times10^{-11}\times100\times10^{4}=1.77\times10^{-5}\ \text{A}$$.

Answer: $$1.8\times10^{-5}\ \text{A}$$.

For additional practice of similar capacitor-based numericals you can browse the solved archive under JEE Questions.

Speed of electromagnetic waves in a material

$$v=\dfrac{1}{\sqrt{\mu\varepsilon}}.$$

• For free space $$\mu_0=4\pi\times10^{-7}\ \text{H m}^{-1},\;\varepsilon_0=8.85\times10^{-12}\ \text{F m}^{-1}$$ ⇒ $$c=3.00\times10^{8}\ \text{m s}^{-1}$$.
• Relative speed $$v=\dfrac{c}{\sqrt{\mu_r\varepsilon_r}}=\dfrac{c}{n}$$ when $$\mu_r\approx1$$ (non-magnetic media).

Energy, Momentum and Radiation Pressure of EM Waves

Energy density

Electric part: $$u_E=\dfrac{1}{2}\varepsilon_0 E^2$$, magnetic part: $$u_B=\dfrac{1}{2}\dfrac{B^2}{\mu_0}$$. For a plane wave $$u_E=u_B$$, so total $$u=u_E+u_B=\varepsilon_0 E^2=\dfrac{B^2}{\mu_0}$$.

Poynting vector

$$\mathbf S=\dfrac{1}{\mu_0}\mathbf E\times \mathbf B$$

|$$\mathbf S$$| equals instantaneous power per unit area (W m^-2). For a sinusoidal wave the average value over one cycle is

$$\langle S\rangle =\dfrac{1}{2}\varepsilon_0 c E_0^2=\dfrac{1}{2}\dfrac{c}{\mu_0}B_0^2.$$

Intensity

In JEE problems intensity is almost always the time-average Poynting magnitude. Remember $$I\propto E_0^2$$.

Momentum of radiation

• Energy flux $$\langle S\rangle$$ corresponds to momentum flux $$\dfrac{\langle S\rangle}{c}$$.
• Radiation pressure $$P_{\rm rad}=\dfrac{I}{c}$$ for complete absorption and $$P_{\rm rad}=\dfrac{2I}{c}$$ for perfect reflection.

Worked example: Solar sail

A reflective sail of area $$50\ \text{m}^2$$ is placed perpendicular to sunlight of intensity $$I=1.36\times10^{3}\ \text{W m}^{-2}$$.

Find the force on the sail.

Radiation pressure $$P=\dfrac{2I}{c}= \dfrac{2(1.36\times10^{3})}{3.00\times10^{8}}=9.07\times10^{-6}\ \text{N m}^{-2}$$.

Force $$F=PA=9.07\times10^{-6}\times50=4.54\times10^{-4}\ \text{N}$$.

Answer: $$4.5\times10^{-4}\ \text{N}$$.

If you struggle with Poynting-vector algebra review the PDF inside our JEE Formula Sheets – it lists every step in one page.

Electromagnetic Spectrum and Practical Sources

Regions to remember

Detecting wavelength in the lab

Expect JEE to ask for order of magnitude identify-the-region or the correct source/detector pairing. Learn the one most distinctive fact for each band, e.g. microwaves use magnetron/klystron.

Need structured practice across the spectrum? The live batch inside JEE Mains Online Coaching includes a micro-quiz after every lecture so you memorise the list mechanically.

Important Formulas and Results at a Glance

$$c=\dfrac{1}{\sqrt{\mu_0\varepsilon_0}}=3.00\times10^{8}\ \text{m s}^{-1},\qquad \varepsilon_0=8.85\times 10^{-12}\ \text{F m}^{-1},\qquad \mu_0=4\pi\times10^{-7}\ \text{H m}^{-1}.$$

Print this table and tape it above your study desk. After you can reproduce every line without looking, switch to mixed-topic problems from JEE Mains Previous Papers to check retention.

JEE Important Points, Common Mistakes and Quick Revision

• Phase lag confusion: Electric and magnetic fields are not out of phase; they reach maxima simultaneously. Many objective options hide this trap.
• Direction of $$\mathbf B$$ in a plane wave: Use right-hand rule $$\mathbf E\times \mathbf B$$; reverse sign if travelling in negative x direction.
• Capacitor displacement current: In a charging capacitor $$I_C = I_D$$ always. Do not zero the magnetic field just because the region is “vacuum”.
• Radiation pressure factor 2: For reflection, momentum change doubles. JEE loves to flip absorption vs reflection quietly in the last line.
• Spectrum order: Frequency increases Radio → Gamma. Learn three anchor wavelengths: 1 m (end of radio), 700 nm (red light), 0.1 nm (hard X-ray).
• Intrinsic impedance: $$Z_0\approx377\ \Omega$$ could appear as a numerical match option.
• Units check: Energy density in J m^-3, intensity in W m^-2, pressure in N m^-2 – dividing intensity by c gives correct pressure dimension.
• Advanced-level links: Combined questions may tie Poynting vector to RLC antenna length. Keep antenna fundamentals from the Communication chapter handy.

Once you review these pitfalls, rapid-fire 20 MCQs in a row. If you need a graded set that escalates, download the booklet in JEE Advanced Previous Papers and filter to Electromagnetic Waves.