Wavelengths of different radiations are given below : λ(A) = 300 nm λ(B) = 300 μm λ(C) = 3 nm λ(D) = 30 Å Arra

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Q. Wavelengths of different radiations are given below : λ(A) = 300 nm λ(B) = 300 μm λ(C) = 3 nm λ(D) = 30 Å Arrange these radiations in the increasing order of their energies.

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Concept understanding — Electromagnetic Wave Wavelength

What is Wavelength? The Intuition First

Imagine dropping a pebble into a still pond. Ripples spread outward — evenly spaced circles. The distance between two consecutive crests (the highest points) is what we call wavelength. It's the "repeat length" of the wave.

Now, electromagnetic waves are not water waves. They don't need a medium. But the same idea holds: an EM wave is a travelling disturbance in electric and magnetic fields. As it moves, the fields oscillate — they go up, down, up, down. The wavelength ( $\lambda$ , Greek letter lambda) is the distance over which the wave's shape repeats.

Note

In a vacuum, all EM waves travel at the same speed: $c = 3 \times 10^8 \ \text{m/s}$ . What changes from one wave to another is the wavelength (and its partner, frequency).

The Precise Definition

For a sinusoidal electromagnetic wave travelling in one direction, the electric field at a fixed instant of time looks like a sine curve in space. The wavelength $\lambda$ is the spatial distance between two successive points that are in phase — for example, from one crest to the next crest, or from one trough to the next trough.

Mathematically, if the electric field at position $x$ and time $t$ is given by

E(x,t) = E_0 \sin(kx - \omega t)

then the wave number $k$ is related to wavelength by

k = \frac{2\pi}{\lambda}

So $\lambda$ is the distance needed for the argument $kx$ to change by $2\pi$ — one full cycle of the sine wave.

The Fundamental Relationship

The wavelength, frequency $f$ , and speed $c$ are tied together by a simple equation:

c = f \lambda

This is the wave equation for EM waves in vacuum. It means:

If the wavelength is long, the frequency is low.
If the wavelength is short, the frequency is high.
The product is always $c$ , a constant.

Tip

To remember: think of a marching band. If soldiers take long strides (large $\lambda$ ), they take fewer steps per second (low $f$ ). If they take short, quick steps (small $\lambda$ ), they take many steps per second (high $f$ ). The speed of the band is fixed — stride length × steps per second.

The Electromagnetic Spectrum

Wavelength is what separates different kinds of EM radiation. Here's the spectrum from longest to shortest wavelength:

Type of EM Wave	Approximate Wavelength Range
Radio waves	$> 0.1 \ \text{m}$ (up to km)
Microwaves	$1 \ \text{mm}$ to $0.1 \ \text{m}$
Infrared	$700 \ \text{nm}$ to $1 \ \text{mm}$
Visible light	$400 \ \text{nm}$ to $700 \ \text{nm}$
Ultraviolet	$10 \ \text{nm}$ to $400 \ \text{nm}$
X-rays	$0.01 \ \text{nm}$ to $10 \ \text{nm}$
Gamma rays	$< 0.01 \ \text{nm}$

Watch out

A common mistake: thinking that wavelength is the "size" of the wave. It's not. It's the repeat distance. A radio wave can have a wavelength of 1 km, but its amplitude (the strength of the field) might be tiny. Wavelength and amplitude are independent properties.

Why Wavelength Matters …

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The energy of an electromagnetic wave is inversely proportional to its wavelength. Converting all wavelengths to meters and comparing gives the increasing order of energy: B < A < D < C.

The key idea here is the relationship between the energy of a photon and its wavelength. For electromagnetic radiation, each photon carries energy given by $E = \frac{hc}{\lambda}$ , where $h$ is Planck's constant and $c$ is the speed of light. Since $h$ and $c$ are constants, energy is inversely proportional to wavelength — shorter wavelength means higher energy, and longer wavelength means lower energy.

To compare the energies, we first need all wavelengths in the same unit. Let's convert each to meters.

Convert all wavelengths to meters
- λ(A) = 300 nm = $300 \times 10^{-9}$ m = $3.00 \times 10^{-7}$ m
- λ(B) = 300 μm = $300 \times 10^{-6}$ m = $3.00 \times 10^{-4}$ m
- λ(C) = 3 nm = $3 \times 10^{-9}$ m = $3.00 \times 10^{-9}$ m
- λ(D) = 30 Å = $30 \times 10^{-10}$ m = $3.00 \times 10^{-9}$ m
Notice that λ(C) and λ(D) are actually equal: 3 nm = 30 Å (since 1 Å = $10^{-10}$ m and 1 nm = $10^{-9}$ m). So C and D have the same wavelength.
Arrange wavelengths in increasing order

Smaller wavelength → higher energy. So we sort from smallest to largest:
- λ(C) = λ(D) = $3.00 \times 10^{-9}$ m (smallest)
- λ(A) = $3.00 \times 10^{-7}$ m
- λ(B) = $3.00 \times 10^{-4}$ m (largest)
Translate to increasing order of energy

Since energy is inversely proportional to wavelength, the increasing order of energy is the reverse of the increasing order of wavelength:
- Lowest energy: longest wavelength → B
- Next: A …

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Q.Wavelengths of different radiations are given below : λ(A) = 300 nm λ(B) = 300 μm λ(C) = 3 nm λ(D) = 30 Å Arrange these radiations in the increasing order of their energies.

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Q.Wavelengths of different radiations are given below :
λ(A) = 300 nm λ(B) = 300 μm λ(C) = 3 nm λ(D) = 30 Å
Arrange these radiations in the increasing order of their energies.

Unlock everything free for 14 days