📐 Formula Bank & Dimensional Analysis

Complete formula collection with derivations and dimensions

JEE doesn't test memorization alone. You must understand when to use which formula, why it works, and how to derive it if needed.

Fundamental Wave Relations

1. Wave Speed in Vacuum

c = νλ = 3 × 10⁸ m/s

Symbols:

c = speed of light in vacuum (m/s)
ν (nu) = frequency (Hz)
λ (lambda) = wavelength (m)

This is universal for ALL EM waves in vacuum. Doesn't matter if it's radio or gamma ray—speed is always c.

Dimensional formula:

[c] = [LT⁻¹]

2. Speed from ε₀ and μ₀

c = 1/√(μ₀ε₀)

Constants:

μ₀ = 4π × 10⁻⁷ T·m/A (permeability of free space)
ε₀ = 8.85 × 10⁻¹² C²/N·m² (permittivity of free space)

This formula connects electromagnetism to wave propagation. Maxwell derived this and realized it matched the known speed of light, proving light is an EM wave!

3. Speed in Medium

v = c/n = 1/√(με)

Symbols:

v = speed in medium
n = refractive index
μ = permeability of medium
ε = permittivity of medium

When light enters a medium: frequency ν stays constant, but wavelength λ and speed v both decrease.

4. E and B Relation

E/B = c (in vacuum)

E₀/B₀ = c

Meaning: At any instant, ratio of electric to magnetic field equals wave speed.

If E₀ = 600 V/m, then B₀ = 600/(3×10⁸) = 2×10⁻⁶ T = 2 μT

5. Wave Equations

E = E₀ sin(kx - ωt)
B = B₀ sin(kx - ωt)

Symbols:

E₀, B₀ = amplitudes
k = wave number = 2π/λ
ω = angular frequency = 2πν

Wave speed: c = ω/k = νλ

6. Photon Energy

E = hν = hc/λ

Planck's constant: h = 6.626 × 10⁻³⁴ J·s

E (in eV) = 1240/λ(nm)

This formula connects wave and particle nature. Higher frequency → higher energy. This is crucial for photoelectric effect and modern physics.

Maxwell's Equations - Formula Form

1. Gauss's Law (Electric)

∮ E · dA = Q/ε₀

Or: ∇ · E = ρ/ε₀

Meaning: Electric flux through closed surface = charge enclosed / ε₀

Dimensions: [ML³T⁻³A⁻¹] = [ML³T⁻³A⁻¹]

2. Gauss's Law (Magnetic)

∮ B · dA = 0

Or: ∇ · B = 0

Meaning: No magnetic monopoles exist. Magnetic field lines form closed loops.

3. Faraday's Law

∮ E · dl = -dΦ_B/dt

Or: ∇ × E = -∂B/∂t

Meaning: Changing magnetic field creates electric field.

ε = -N(dΦ_B/dt)

This is the basis of EM induction.

4. Ampere-Maxwell Law

∮ B · dl = μ₀I + μ₀ε₀(dΦ_E/dt)

Or: ∇ × B = μ₀J + μ₀ε₀(∂E/∂t)

Displacement current:

I_d = ε₀(dΦ_E/dt)

Meaning: Changing electric field (displacement current) creates magnetic field, just like real current does.

Maxwell's addition of the displacement current term made the equations symmetric and predicted EM waves!

Energy, Intensity, and Momentum Formulas

Energy Density

u_E = (1/2)ε₀E² (electric)
u_B = (1/2μ₀)B² (magnetic)
u = u_E + u_B = ε₀E² = B²/μ₀ (total)

u_E = u_B always! Electric and magnetic fields carry equal energy in EM waves.

Dimensional formula: [u] = [ML⁻¹T⁻²]

Intensity (Poynting Vector)

S = (1/μ₀)(E × B) (vector form)
I = |S| = (EB)/μ₀ = cε₀E² = cB²/μ₀

Average intensity:

I_avg = (1/2)cε₀E₀² = (c/2μ₀)B₀²

I_avg = c × u_avg

Dimensions: [I] = [MT⁻³]

Intensity is power per unit area. For spherical wave: I ∝ 1/r² (inverse square law).

Momentum Relations

p = U/c

where U is energy of EM wave.

Momentum density = u/c

For photon:

p = E/c = hν/c = h/λ

Radiation Pressure

P = I/c (complete absorption)
P = 2I/c (complete reflection)

Or in terms of energy density:

P = u (absorption)
P = 2u (reflection)

Force = Pressure × Area

Dimensions: [P] = [ML⁻¹T⁻²]

Pressure doubles for reflection because momentum change is 2p (like elastic collision), not p.

Malus's Law (Polarization)

I = I₀ cos²θ

where θ is angle between polarizer axis and polarization direction.

For unpolarized light through polarizer:

I = I₀/2

EM Wave Calculator

Wavelength Calculator

Frequency (Hz):

Photon Energy Calculator

Frequency (Hz):

EM Spectrum Quick Reference

Type	Frequency Range	Wavelength Range
Radio	< 10⁹ Hz	> 0.1 m
Microwave	10⁹ - 10¹² Hz	1 mm - 0.1 m
Infrared	10¹² - 4×10¹⁴ Hz	700 nm - 1 mm
Visible	4×10¹⁴ - 7.5×10¹⁴ Hz	400 - 700 nm
UV	7.5×10¹⁴ - 10¹⁷ Hz	10 - 400 nm
X-ray	10¹⁷ - 10²⁰ Hz	0.01 - 10 nm
Gamma	> 10²⁰ Hz	< 0.01 nm

Dimensional Analysis Checklist

Important Dimensions

Speed (c, v)	[LT⁻¹]
Electric field (E)	[MLT⁻³A⁻¹]
Magnetic field (B)	[MT⁻²A⁻¹]
Energy density (u)	[ML⁻¹T⁻²]
Intensity (I)	[MT⁻³]
Pressure (P)	[ML⁻¹T⁻²]

Verification Tips

Always check dimensions of final answer
E/B must have dimension of velocity
Energy density and pressure have same dimensions
Intensity = power/area = energy/(area × time)
Use dimensional analysis to remember formulas

Next: Wave Properties →