Electromagnetic Waves formulas for JEE Main & Advanced

16 formula entries for Electromagnetic Waves, in the order the chapter builds them. Each shows the formula, when to use it and the first trap to avoid.

Maxwell's Equations (Integral Form)

\oint \vec{E} \cdot d \vec{A} = \frac{Q_{enc}}{ϵ_{0}}

\oint \vec{B} \cdot d \vec{A} = 0

\oint \vec{E} \cdot d \vec{l} = - \frac{d Φ_{B}}{d t}

\oint \vec{B} \cdot d \vec{l} = μ_{0} I_{enc} + μ_{0} ϵ_{0} \frac{d Φ_{E}}{d t}

When to use. When deriving wave equations or understanding the origin of EM waves.

Trap. Forgetting the displacement current term in Ampere-Maxwell law.

Displacement Current

I_{d} = ϵ_{0} \frac{d Φ_{E}}{d t}

When to use. When analyzing circuits with capacitors or EM wave generation.

Trap. Thinking displacement current is a real current; it is a concept to maintain continuity.

Speed of EM Waves in Vacuum

c = \frac{1}{\sqrt{μ_{0} ϵ_{0}}} = 3 \times 10^{8} m/s

When to use. When calculating wavelength, frequency, or time of flight for EM waves.

Trap. Using $c$ as speed in a medium; it is only in vacuum.

Relation Between E and B in EM Wave

E_{0} = c B_{0}

E = c B

(instantaneous)

When to use. When given one field amplitude to find the other.

Trap. Using $E = c B$ in a medium; it holds only in vacuum.

Energy Density of EM Wave

u = \frac{1}{2} ϵ_{0} E^{2} + \frac{1}{2} \frac{B^{2}}{μ_{0}} = ϵ_{0} E^{2} = \frac{B^{2}}{μ_{0}}

(in vacuum)

When to use. When calculating energy stored in an EM wave or comparing fields.

Trap. Forgetting that in vacuum, electric and magnetic energy densities are equal.

Intensity of EM Wave

I = \frac{P}{A} = \frac{1}{2} ϵ_{0} c E_{0}^{2} = \frac{1}{2} \frac{c}{μ_{0}} B_{0}^{2} = \frac{E_{0}^{2}}{2 μ_{0} c}

When to use. When calculating power radiated by a source or received by a detector.

Trap. Using RMS values instead of peak values in the formula; $I = \frac{1}{2} ϵ_{0} c E_{0}^{2}$ uses peak E.

Poynting Vector

\vec{S} = \frac{1}{μ_{0}} \vec{E} \times \vec{B}

Magnitude:

S = \frac{E B}{μ_{0}}

When to use. When finding direction of energy propagation or instantaneous power flow.

Trap. Forgetting the cross product order; it's $\vec{E} \times \vec{B}$ , not $\vec{B} \times \vec{E}$ .

Electromagnetic Spectrum

c = f λ

When to use. When identifying wave type based on frequency or wavelength.

Trap. Confusing the order: increasing frequency: radio, microwave, IR, visible, UV, X-ray, gamma.

Solving EM Wave Problems Using Wave Equation

k = \frac{2 π}{λ}

ω = 2 π f

c = \frac{ω}{k}

When to use. When given a sinusoidal EM wave and asked to find frequency, wavelength, or field values.

Trap. Forgetting that E and B are in phase in vacuum.

Calculating Displacement Current in Capacitor

I_{d} = ϵ_{0} \frac{d Φ_{E}}{d t} = ϵ_{0} A \frac{d E}{d t}

(for parallel plate capacitor)

When to use. When analyzing current in capacitor circuit or EM wave generation.

Trap. Forgetting that $Φ_{E} = E A$ only if E is uniform.

Finding Radiation Pressure

For perfect absorption:

P = \frac{I}{c}

For perfect reflection:

P = \frac{2 I}{c}

When to use. When calculating force on a surface due to EM wave.

Trap. Using $P = I / c$ for reflection; it's double.

Speed of EM Wave in a Medium

v = \frac{1}{\sqrt{μ ϵ}} = \frac{c}{n}

where

n = \sqrt{μ_{r} ϵ_{r}}

is refractive index

When to use. When calculating speed, wavelength, or frequency in a medium.

Trap. Using $v = c / n$ only for non-magnetic materials; $μ_{r} \approx 1$ for most.

Calculating Energy and Momentum of EM Wave

Momentum density:

\frac{u}{c}

Momentum per unit area per time:

\frac{I}{c}

(absorption)

When to use. When calculating force or momentum transfer from EM waves.

Trap. Forgetting that momentum is a vector; direction same as propagation.

Average Intensity from RMS Fields

I = ϵ_{0} c E_{rms}^{2} = \frac{c}{μ_{0}} B_{rms}^{2}

where

E_{rms} = \frac{E_{0}}{\sqrt{2}}

B_{rms} = \frac{B_{0}}{\sqrt{2}}

When to use. When given RMS values of fields to find intensity.

Trap. Using peak values in RMS formula.

Solving Problems with EM Wave in Dielectric

v = \frac{c}{n}

λ_{medium} = \frac{λ_{0}}{n}

f

unchanged

When to use. When EM wave enters a medium from vacuum.

Trap. Thinking frequency changes in medium; it does not.

Doppler Effect in EM Waves

f^{'} = f \sqrt{\frac{1 \pm β}{1 \mp β}}

where

β = v / c

(relativistic formula)

When to use. When calculating frequency shift due to relative motion.

Trap. Using classical Doppler formula for sound; EM waves require relativistic formula.

Values to remember

$μ_{0} = 4 π \times 10^{- 7} H/m$ Speed of EM Waves in Vacuum
$ϵ_{0} = 8.85 \times 10^{- 12} F/m$ Speed of EM Waves in Vacuum
$c = 3 \times 10^{8} m/s$ Speed of EM Waves in Vacuum
Visible: 400-700 nm Electromagnetic Spectrum
Radio: > 0.1 m Electromagnetic Spectrum
Microwave: 1 mm - 0.1 m Electromagnetic Spectrum
IR: 700 nm - 1 mm Electromagnetic Spectrum
UV: 10-400 nm Electromagnetic Spectrum
X-ray: 0.01-10 nm Electromagnetic Spectrum
Gamma: < 0.01 nm Electromagnetic Spectrum

Read the full chapter: Electromagnetic Waves for JEE, with what JEE asks and the common traps.