A hypothetical electromagnetic wave is shown in the figure below. Find out the wavelength of the radiation. ![A hypothetical electromagnetic wave: a sinusoidal wave on a horizontal axis, with a marked horizontal distance of 2.16 pm between the first crest and the immediately following point where the wave crosses the axis](/figures/structure-of-atom/hypothetical-electromagnetic-wave-quarter-wavelength-exemplar-fig-2-2.svg)

> [!TLDR] > The distance from a crest to an immediately following node in a sinusoidal wave represents one-quarter of its full wavelength. Given this distance as 2.16 pm, the wavelength is $\boxed{8.64 \text{ pm}}$. Electromagnetic waves, like all waves, exhibit a periodic pattern. When we represent them as a smooth sinusoidal wave, certain points on the wave have specific names and relate directly to the wave's fundamental length scale: its wavelength. Understanding these basic anatomical features of a wave is crucial for solving problems involving wave properties. 1. **Understanding Wave Anatomy:** A sinusoidal wave oscillates around a central equilibrium position (the horizontal axis in this case). * A **crest** is a point on the wave where its displacement from the equilibrium position is maximum in the positive direction. * A **trough** is a point where its displacement is maximum in the negative direction. * A **node** (or zero-crossing point) is a point where the wave crosses the equilibrium axis, meaning its displacement is zero. * The **wavelength** ($\lambda$) is the spatial period of the wave, which is the distance over which the wave's shape repeats. It can be measured, for example, from one crest to the next adjacent crest, or from one trough to the next adjacent trough. 2. **Relating Crest-to-Node Distance to Wavelength:** Consider a full cycle of a sinusoidal wave. * From a crest to the next trough is half a wavelength ($\lambda/2$). * From a crest to the immediately following node (where the wave crosses the axis going downwards) is exactly one-quarter of a wavelength ($\lambda/4$). * Similarly, from a node to the immediately following crest (where the wave crosses the axis going upwards) is also one-quarter of a wavelength ($\lambda/4$). …

A hypothetical electromagnetic wave is shown in the figure below. Find out the wavelength of the radiation.

NCERT Exemplar · Q36

Q. A hypothetical electromagnetic wave is shown in the figure below. Find out the wavelength of the radiation.

A hypothetical electromagnetic wave: a sinusoidal wave on a horizontal axis, with a marked horizontal distance of 2.16 pm between the first crest and the immediately following point where the wave crosses the axis — Figure 2.2

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Concept understanding — Energy Level Quantization

Energy Level Quantization: From Intuition to Precision

Imagine you're climbing a smooth ramp. You can stop at any height — 1 metre, 1.5 metres, 2.1 metres — anywhere you like. That's how we intuitively think about energy in everyday life: continuous, like a slide.

Now imagine a staircase. You can stand on step 1, step 2, or step 3 — but you cannot stand halfway between step 2 and step 3. There's no such place. The steps are discrete, not continuous.

Energy level quantization is the idea that in the microscopic world of atoms and molecules, energy behaves like a staircase, not a ramp. Electrons in an atom cannot have just any energy — they can only occupy specific, allowed energy levels. Everything else is forbidden.

Why does this happen? The core intuition

In the classical world, an electron orbiting a nucleus would continuously radiate energy, spiral inward, and crash — atoms would be unstable. But atoms are stable. Nature solved this problem by imposing a rule: the electron's angular momentum (and therefore its energy) can only take certain discrete values.

Think of a guitar string. It can only vibrate at specific frequencies — its fundamental and harmonics. You can't pluck it to produce a frequency halfway between two harmonics. The string's vibration is quantized by its boundaries. Similarly, an electron bound to a nucleus is confined in space, and that confinement forces its energy to be quantized.

Note

Quantization is not a mysterious extra rule — it emerges naturally whenever a wave (like an electron's matter wave) is confined. Confinement creates standing waves, and standing waves only exist at specific frequencies.

The precise statement

For a bound system (like an electron in an atom), the total energy $E$ of the system can only take certain discrete values:

E = E_1, E_2, E_3, \dots

where each $E_n$ is a specific, fixed number. The integer $n$ (1, 2, 3, …) is called the principal quantum number. The lowest energy level ( $n=1$ ) is the ground state; higher levels ( $n>1$ ) are excited states.

For the hydrogen atom, the allowed energies are given by:

E_n = -\frac{13.6 \text{ eV}}{n^2}

So:

$n=1$ : $E_1 = -13.6$ eV (ground state)
$n=2$ : $E_2 = -3.4$ eV
$n=3$ : $E_3 = -1.51$ eV
and so on, approaching 0 eV as $n \to \infty$ (the ionization limit)

Important

The negative sign means the electron is bound to the nucleus. Zero energy corresponds to the electron being free (ionized). The more negative the energy, the more tightly bound the electron.

How do we know this is real?

The most direct evidence comes from atomic spectra. When an electron jumps from a higher energy level to a lower one, it emits a photon of light with energy exactly equal to the difference:

\Delta E = E_{\text{higher}} - E_{\text{lower}} = hf

where $h$ is Planck's constant and $f$ is the frequency of the emitted light. …

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The distance from a crest to an immediately following node in a sinusoidal wave represents one-quarter of its full wavelength. Given this distance as 2.16 pm, the wavelength is $\boxed{8.64 \text{ pm}}$ .

Electromagnetic waves, like all waves, exhibit a periodic pattern. When we represent them as a smooth sinusoidal wave, certain points on the wave have specific names and relate directly to the wave's fundamental length scale: its wavelength. Understanding these basic anatomical features of a wave is crucial for solving problems involving wave properties.

Understanding Wave Anatomy:

A sinusoidal wave oscillates around a central equilibrium position (the horizontal axis in this case).
- A crest is a point on the wave where its displacement from the equilibrium position is maximum in the positive direction.
- A trough is a point where its displacement is maximum in the negative direction.
- A node (or zero-crossing point) is a point where the wave crosses the equilibrium axis, meaning its displacement is zero.
- The wavelength ( $\lambda$ ) is the spatial period of the wave, which is the distance over which the wave's shape repeats. It can be measured, for example, from one crest to the next adjacent crest, or from one trough to the next adjacent trough.
Relating Crest-to-Node Distance to Wavelength:

Consider a full cycle of a sinusoidal wave.
- From a crest to the next trough is half a wavelength ( $\lambda/2$ ).
- From a crest to the immediately following node (where the wave crosses the axis going downwards) is exactly one-quarter of a wavelength ( $\lambda/4$ ).
- Similarly, from a node to the immediately following crest (where the wave crosses the axis going upwards) is also one-quarter of a wavelength ( $\lambda/4$ ). …

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Q.A hypothetical electromagnetic wave is shown in the figure below. Find out the wavelength of the radiation.

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