Which one among the following transitions of electron of hydrogen atom emits radiation of the shortest wavelength?

1. Bohr's Model of the Atom and Energy Levels (basic)

To understand the modern atom, we must start with Niels Bohr’s Model, which revolutionized our view of the microscopic world. Before Bohr, scientists couldn't explain why electrons didn't simply lose energy and spiral into the nucleus. Bohr proposed that electrons do not just float randomly; they revolve around the nucleus in fixed, circular paths called orbits or stationary shells. Think of these shells like the planets orbiting the Sun, but with a twist: the electron can only exist in these specific orbits and nowhere in between.

These orbits are associated with fixed amounts of energy, which is why we call them energy levels. They are labeled using the principal quantum number (n) or letters: n = 1 (K shell), n = 2 (L shell), n = 3 (M shell), and so on. The shell closest to the nucleus (K shell) has the lowest energy, and as we move outward to the L or M shells, the energy level increases Science, Class X (NCERT 2025 ed.), Carbon and its Compounds, p.59. For instance, a hydrogen atom typically has its single electron in the K shell, which is its most stable state Science, Class VIII, NCERT (Revised ed 2025), Particulate Nature of Matter, p.115.

The most critical part of Bohr's theory is quantization. An electron only changes its energy level by "jumping" from one orbit to another.

• Absorption: To move from a lower level (near the nucleus) to a higher level (further away), an electron must absorb a specific amount of energy.
• Emission: When an electron falls from a higher energy level to a lower one, it releases energy in the form of radiation (a photon).

This energy release is what creates the distinct spectral lines we see in physics; the larger the "fall" or jump, the more energy the emitted light carries.

Feature	Inner Shells (e.g., K)	Outer Shells (e.g., M, N)
Distance from Nucleus	Closest	Farther away
Energy Level	Lowest (Ground State)	Higher (Excited States)
Stability	Most Stable	Less Stable

Sources: Science, Class X (NCERT 2025 ed.), Carbon and its Compounds, p.59; Science, Class VIII, NCERT (Revised ed 2025), Particulate Nature of Matter, p.115

2. Wavelength, Frequency, and Energy Relationship (basic)

To understand the physics of atoms, we must first master the language of light and radiation. Every form of electromagnetic radiation—from the radio waves that carry signals to your phone to the UV rays that cause sunburns—is defined by three interconnected properties: wavelength (λ), frequency (ν), and energy (E). The speed of light (c) is a constant in a vacuum, which creates a fundamental trade-off: if the wavelength is long, the frequency must be low, and vice-versa. As noted in Physical Geography by PMF IAS, Earths Atmosphere, p.279, wavelength is inversely proportional to the frequency of the wave. This is why Radio waves have the longest wavelengths, ranging from the size of a football to larger than our planet, while having very low frequencies.

The most critical relationship for a UPSC aspirant to grasp is how these properties determine Energy. According to Planck’s Law (E = hν), energy is directly proportional to frequency. This means that "fast-vibrating" high-frequency waves carry more punch (energy) than "slow-vibrating" low-frequency waves. Consequently, energy is inversely proportional to wavelength. You can see this in how our atmosphere behaves: the Sun, being incredibly hot, emits high-energy short-wave radiation (insolation), while the much cooler Earth emits lower-energy long-wave radiation FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Solar Radiation, Heat Balance and Temperature, p.67.

This relationship also explains why the sky is blue. In the visible spectrum, blue light has a shorter wavelength and higher frequency than red light (red light's wavelength is about 1.8 times greater than blue). Because blue light has a shorter wavelength, it is scattered much more strongly by the fine particles in our atmosphere, filling our vision with a blue hue Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.169. In the context of atomic physics, when an electron jumps between energy levels, it releases a photon. A large energy jump results in a high-energy photon, which corresponds to a short wavelength (like UV light), while a small jump results in a long wavelength (like Infrared).

Property	High Energy Wave	Low Energy Wave
Frequency	High (High ν)	Low (Low ν)
Wavelength	Short (Small λ)	Long (Large λ)
Example	Gamma Rays / Blue Light	Radio Waves / Red Light

Sources: Physical Geography by PMF IAS, Earths Atmosphere, p.279; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Geography Class XI (NCERT 2025 ed.), Solar Radiation, Heat Balance and Temperature, p.67; Science, class X (NCERT 2025 ed.), The Human Eye and the Colourful World, p.169

3. The Electromagnetic Spectrum in Science (intermediate)

To understand atomic transitions, we must first master the Electromagnetic (EM) Spectrum. Think of the EM spectrum as a continuous range of energy traveling as waves. These waves are made of oscillating electric and magnetic fields. While they all travel at the speed of light (c ≈ 3 × 10⁸ m/s) in a vacuum, they differ fundamentally in their wavelength and frequency.

Let's define our terms clearly: Wavelength (λ) is the horizontal distance between two successive crests, while Frequency (f) is the number of waves passing a point in one second Physical Geography by PMF IAS, Tsunami, p.192. There is a crucial inverse relationship here: Energy is directly proportional to frequency, but inversely proportional to wavelength. This means that a wave with a very short wavelength (like a Gamma ray) packs a much more powerful energetic punch than a wave with a long wavelength (like a Radio wave).

Region	Wavelength Trend	Energy/Frequency Trend	Common Interaction
Radio Waves	Longest	Lowest	Reflected by the ionosphere for communication Physical Geography by PMF IAS, Earths Atmosphere, p.279.
Infrared (IR)	Medium-Long	Low	Experienced as heat; absorbed by water vapor and ozone in the troposphere FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Solar Radiation, Heat Balance and Temperature, p.68.
Visible Light	Intermediate	Medium	The only part we see; scattered by particles to create blue skies and red sunsets FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Solar Radiation, Heat Balance and Temperature, p.68.
Ultraviolet (UV)	Short	High	Absorbed by the ozone layer; can be harmful to biological life Environment and Ecology, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8.
X-rays/Gamma	Shortest	Highest	Highly penetrative; often associated with nuclear reactions.

In the context of physics, when an atom releases energy, it does so by emitting a photon (a packet of light). The amount of energy lost by the atom determines exactly where on this spectrum that photon will fall. If the energy release is massive, the wavelength will be very short (UV or X-ray); if the energy release is small, the wavelength will be longer (Infrared or Radio).

Sources: Physical Geography by PMF IAS, Tsunami, p.192; Physical Geography by PMF IAS, Earths Atmosphere, p.279; FUNDAMENTALS OF PHYSICAL GEOGRAPHY, Solar Radiation, Heat Balance and Temperature, p.68; Environment and Ecology, BASIC CONCEPTS OF ENVIRONMENT AND ECOLOGY, p.8

4. Radioactivity and Nuclear Emissions (intermediate)

At its heart, radioactivity is a process of nuclear stabilization. While most atoms in nature are stable, certain elements possess 'unstable' nuclei. To reach a more stable state, these nuclei spontaneously disintegrate, releasing energy and particles in the process Environment, Shankar IAS Academy, Environmental Pollution, p.82. This isn't just a theoretical concept; we harness this by altering the structure of atoms to release immense heat, which is then used to generate electricity in nuclear power plants NCERT, Contemporary India II, Print Culture and the Modern World, p.117. There are three primary types of nuclear emissions you should know for your preparation. Each has distinct physical properties and levels of penetration:

Emission Type	Nature	Description
Alpha (α)	Protons	Heavy, positively charged particles; low penetration power (stopped by paper).
Beta (β)	Electrons	Light, negatively charged particles; moderate penetration.
Gamma (γ)	EM Waves	Short-wave electromagnetic radiation; highly energetic and deeply penetrating.

In the Indian context, our nuclear program relies on specific minerals. Uranium and Thorium are the primary fuels. You will find Uranium in Jharkhand and the Aravalli ranges of Rajasthan, while the Monazite sands of Kerala are a world-class source of Thorium NCERT, Contemporary India II, Print Culture and the Modern World, p.117. To manage these resources, India established a robust institutional framework: However, radioactivity is a double-edged sword. While it provides clean energy, ionizing radiation poses significant biological risks. High doses can damage bone marrow, retarding the body's ability to fight infection, and may lead to long-term health issues like leukemia, bone cancer, or hereditary genetic mutations Environment and Ecology, Majid Hussain, Environmental Degradation and Management, p.44. This is why nuclear safety and waste management remain central themes in environmental governance.

Sources: Environment, Shankar IAS Academy, Environmental Pollution, p.82; NCERT, Contemporary India II, Print Culture and the Modern World, p.117; INDIA PEOPLE AND ECONOMY, NCERT, Mineral and Energy Resources, p.61; Environment and Ecology, Majid Hussain, Environmental Degradation and Management, p.44

5. Nuclear Energy and India's Nuclear Program (exam-level)

To understand India's nuclear journey, we must first distinguish between the two ways of releasing energy from the atom: Nuclear Fission (splitting a heavy nucleus like Uranium) and Nuclear Fusion (combining light nuclei like Hydrogen). While fission powers our current reactors, fusion requires extreme temperatures and pressures similar to the sun; such conditions do not exist naturally within the Earth's interior Physical Geography by PMF IAS, Earths Interior, p.59. India's program, envisioned by Dr. Homi J. Bhabha, was designed as a three-stage program to eventually utilize our vast Thorium reserves, moving from pressurized heavy water reactors to fast breeder reactors, and finally to Thorium-based systems.

India's transition to a recognized nuclear power was marked by two major milestones at the Pokhran test range. The first was in 1974, followed by Operation Shakti in May 1998. This second series involved five underground detonations, including fission, fusion (thermonuclear), and sub-kiloton devices, led by figures like Dr. A.P.J. Abdul Kalam A Brief History of Modern India, After Nehru, p.754. On the civilian side, India has rapidly expanded its capacity. From the first unit at Tarapur in 1969 to the massive 1000 MW units at Kudankulam, the focus has now shifted toward indigenous 700 MW reactors to boost domestic industry and economic development Geography of India, Majid Husain, Energy Resources, p.27.

Beyond the science, the Nuclear Doctrine defines how India handles this power. It is built on the principle of strategic restraint, ensuring the world that India’s capabilities are meant for peace and deterrence rather than aggression.

Principle	Description
No First Use (NFU)	Nuclear weapons will only be used in retaliation against a nuclear attack on Indian territory or forces.
Credible Minimum Deterrent	Maintaining a sufficient stockpile to deter adversaries without entering an arms race.
Civilian Control	Retaliatory strikes can only be authorized by the civilian political leadership via the Nuclear Command Authority.
Non-Use	No use of nuclear weapons against non-nuclear-weapon states Indian Polity, M. Laxmikanth(7th ed.), Foreign Policy, p.611.

Sources: Physical Geography by PMF IAS, Earths Interior, p.59; A Brief History of Modern India, After Nehru, p.754; Geography of India, Majid Husain, Energy Resources, p.27; Indian Polity, M. Laxmikanth(7th ed.), Foreign Policy, p.611

6. Electron Transitions and Spectral Series (Lyman, Balmer) (exam-level)

In the atomic world, electrons do not move randomly; they occupy specific, discrete orbits known as quantized energy levels. According to the Bohr model, these levels are designated by the principal quantum number (n = 1, 2, 3...). As we move further from the nucleus, the energy of the electron increases, but the gap between successive energy levels actually decreases. Think of it like a ladder where the rungs get closer together as you climb higher. This distribution is fundamental to understanding how atoms interact with light Science, Class X (NCERT 2025 ed.), Metals and Non-metals, p.47.

When an electron "jumps" from a higher energy level to a lower one, it must shed its excess energy. It does this by emitting a particle of light called a photon. The energy of this photon is exactly equal to the difference in energy between the two levels (ΔE). This relationship is governed by the equation E = hc/λ, where 'h' is Planck's constant, 'c' is the speed of light, and 'λ' (lambda) is the wavelength. Crucially, energy and wavelength are inversely proportional: a larger energy jump produces a photon with a shorter wavelength.

To help scientists categorize these emissions, transitions are grouped into "Series" based on the final destination of the electron:

Series	Final Level (n₁)	Spectral Region	Energy Magnitude
Lyman	n = 1	Ultraviolet (UV)	Highest Energy (Shortest λ)
Balmer	n = 2	Visible Light	Medium Energy
Paschen	n = 3	Infrared (IR)	Lower Energy (Longer λ)

Because the gap between n = 1 and n = 2 is much larger than any other gap in the atom, any transition ending at n = 1 (Lyman series) will always involve more energy and a shorter wavelength than transitions in the Balmer or Paschen series. For example, the transition from n = 2 to n = 1 emits ultraviolet light at approximately 121.6 nm, while the n = 3 to n = 2 transition results in red visible light.

Sources: Science, Class X (NCERT 2025 ed.), Metals and Non-metals, p.47

7. Solving the Original PYQ (exam-level)

To solve this, we must bridge two fundamental concepts you’ve just mastered: the inverse relationship between energy and wavelength (E = hc/λ) and the quantized energy levels of the Bohr model. You learned that as the principal quantum number (n) increases, the energy levels of a hydrogen atom actually get closer together. Therefore, the transition with the largest energy gap will automatically produce the radiation with the shortest wavelength. This is the core logic UPSC expects you to apply under pressure.

Walking through the options, the jump from n = 2 to n = 1 (the first line of the Lyman series) represents the single largest energy transition possible between any two adjacent shells. As you move to higher shells—like n = 3 to n = 2 or n = 5 to n = 4—the "rungs" of the energy ladder become much narrower. Because the energy released in (A) n = 2 to n = 1 is the highest, its wavelength is the shortest, falling into the ultraviolet spectrum, whereas the other options move toward visible or infrared light. Reference: Wikipedia: Hydrogen spectral series.

The common trap here is assuming that "higher" shell numbers (like n = 5 or n = 4) imply "higher" energy transitions. UPSC often uses this numerical intuition trap to see if you understand the actual physical spacing of the atom. While n = 5 is a higher energy state than n = 2, the difference (ΔE) between n = 5 and n = 4 is much smaller than the massive gap between the ground state (n = 1) and the first excited state (n = 2). This is why Option (A) is the only logically sound choice.