Q.The wavelength of a photon needed to remove a proton from a nucleus which is bound to the nucleus with 1 MeV energy is nearly
Concept understanding — Photon Energy
Photon Energy: The Energy Carried by Light
Light does not carry its energy in a continuous stream. It comes in discrete packets — like individual drops instead of a steady hose. Each packet is called a photon: the smallest possible unit of light of a given frequency. You cannot have half a photon; it is all or nothing.
The Intuition: Colour Determines Energy
Red light and blue light are both light, but they behave differently — blue light causes sunburn and drives chemical reactions more readily than red. The reason is that the colour (frequency) of light directly fixes how much energy each photon carries, and blue photons carry more energy than red photons.
The Planck–Einstein Relation
The energy E of a single photon is directly proportional to its frequency f, and inversely proportional to its wavelength λ:
E=hf=λhc
Where:
- E = energy of one photon (joules, J)
- f = frequency (hertz, Hz)
- λ = wavelength (metres, m)
- c = speed of light in vacuum (3.00×108 m/s)
- h = Planck's constant (6.626×10−34 J·s)
Planck's constant h is extremely tiny, so an individual photon carries a very small amount of energy — which is why we never feel single photons striking our skin.
What This Tells You
- Higher frequency = higher energy. Gamma rays have extremely energetic photons; radio waves have very low-energy photons.
- Shorter wavelength = higher energy. Ultraviolet photons are more energetic than visible-light photons; infrared photons are less.
- Energy is quantised. Light energy comes in fixed packets — you can have 1, 2, or 1000 photons, but never 0.5. This was Max Planck's revolutionary idea of 1900.
This is why UV light causes sunburn (UV photons carry enough energy to damage DNA, while visible photons do not), why X-ray photons are energetic enough to pass through soft tissue but get absorbed by bone, and why a photon needs enough energy to actually break a bond or drive a photochemical/biological reaction, rather than merely wavelength-dependent intensity.
A Concrete Example
Question: Which has more energy — a photon of red light (λ=700 nm) or a photon of blue light (λ=450 nm)?
Since E=λhc, energy is inversely proportional to wavelength, so the shorter-wavelength blue photon carries more energy:
- Red: Ered=700×10−9(6.626×10−34)(3.00×108)≈2.84×10−19 J
- Blue: Eblue=450×10−9(6.626×10−34)(3.00×108)≈4.42×10−19 J
The blue photon carries about 1.5 times the energy of the red photon.
A common mistake is to think brighter light means more energetic photons. Brightness is the number of photons per second, not the energy per photon. A bright red light has many low-energy photons; a dim blue light has fewer but higher-energy ones.
The Big Picture
Photon energy is the bridge between the wave nature of light (frequency, wavelength) and its particle nature (energy packets) — one of the foundational ideas of quantum mechanics: at the smallest scales, energy is not continuous but comes in discrete, indivisible units.
Photon energy, given by the Planck-Einstein relation E = hf, is one of the most fundamental formulas in the NCERT Class 12 Physics Dual Nature of Radiation and Matter chapter, and "photon energy formula and calculation" is a heavily searched query among students preparing for CBSE boards, JEE Main, and NEET. Because this idea links directly to the photoelectric effect and atomic spectra, it also anchors several "modern physics important questions" compiled for competitive-exam revision.
Why this formula?
Why Photon Energy is E=hf — The Reasoning Behind the Formula
The formula E=hf is not something you memorise and plug numbers into. It comes from a deep shift in how physicists understood light.
The problem that forced the formula
By the late 1800s, classical physics said light was a wave — continuous, carrying energy in proportion to its intensity (brightness). But blackbody radiation and the photoelectric effect showed something bizarre: when you shine light on a metal, electrons are ejected only if the light's frequency is above a certain threshold, no matter how bright the light is. Below that frequency, even the brightest lamp won't kick out a single electron. Classically this made no sense — a wave's energy depends on amplitude, not frequency.
Einstein's radical idea (1905)
Einstein proposed that light comes in discrete packets — photons — each carrying a fixed energy that depends only on its frequency, with Planck's constant h as the proportionality:
A single photon's energy is E=hf, where f is the frequency of the light.
This directly explains the photoelectric effect: an electron needs a minimum energy (the work function ϕ) to escape. If a photon's energy hf<ϕ, no electron is ejected — regardless of how many photons you send, because each photon interacts with one electron individually.
E=hf is not derived from deeper principles — it is a postulate justified by the experiments it explains, and it unified optics with quantum mechanics.
The equivalent forms
Since c=fλ for light in vacuum, the same idea in terms of wavelength is:
E=λhc
Use this when you're given λ instead of f. A photon also carries momentum (despite having no mass):
p=cE=λh
This follows from special relativity: E2=(pc)2+(mc2)2, and for a photon m=0, so E=pc, hence p=h/λ.
- h=6.626×10−34 J⋅s (Planck's constant)
- c=3.0×108 m/s (speed of light)
- For atomic-scale problems use electronvolts: 1 eV=1.602×10−19 J
A common exam trap
E=hf means higher frequency = more energy per photon — true. But a brighter light does not mean higher photon energy. Intensity is the number of photons per second per area; each photon still carries hf.
Do not confuse intensity (number of photons) with frequency (energy per photon). They are independent.
Final answer: E=hf
The key idea is that the photon must supply at least the binding energy of the proton — here 1 MeV.
Reasoning:
- The minimum photon energy required is E=1 MeV=106 eV.
- Use the photon energy-wavelength relation:
E=λhc⇒λ=Ehc.
- A useful shortcut: hc≈1240 eV⋅nm. So
λ=106 eV1240 eV⋅nm=1.24×10−3 nm.
This matches option (B).
The wavelength is nearly 1.2×10−3 nm, which corresponds to option (B).
The photon must supply exactly the binding energy of the proton (1 MeV). Using E=hc/λ, the wavelength comes out to about 1.24×10−3 nm, which matches option (B).
The core idea here is that removing a proton from a nucleus requires overcoming the nuclear binding force. That binding energy is given as 1 MeV — the minimum energy a photon must carry to eject the proton. Since a photon’s energy is inversely proportional to its wavelength, we can directly compute the wavelength.
A common pitfall is forgetting to convert units properly or mixing up the energy-wavelength relation for photons. Let’s walk through it cleanly.
- Recall the photon energy-wavelength relation For any photon, E=λhc, where h is Planck’s constant and c is the speed of light. The product hc is a very useful constant:
hc=1240 eV⋅nm
(This is exact enough for all exam purposes — it comes from h=4.135667×10−15 eV⋅s and c=2.998×108 m/s, giving hc≈1240 eV⋅nm.)
- Set the photon energy equal to the binding energy The photon must have E=1 MeV=106 eV. So:
λhc=106 eV
- Solve for λ
λ=106 eVhc=106 eV1240 eV⋅nm=1.24×10−3 nm
- Match with the options The value 1.24×10−3 nm is extremely close to 1.2×10−3 nm — the slight difference is due to rounding hc to 1240 instead of 1239.84. In multiple-choice exams, this is the intended match.
A very common mistake is to use E=hf and then forget that c=fλ, or to mix up units (e.g., using hc=1240 eV⋅nm but then treating the energy in MeV without converting to eV). Always convert MeV to eV first: 1 MeV=106 eV.
Memorising hc=1240 eV⋅nm saves enormous time. For any photon energy in eV, the wavelength in nm is simply 1240/E. For MeV energies, just shift the decimal: 1240/106=1.24×10−3.
The correct option is (B) 1.2×10−3 nm.
Method: Converting a Threshold/Binding Energy into a Photon Wavelength
Use this whenever a question gives you a minimum energy a photon must supply (a binding energy, an ionisation energy, a work function) and asks for the corresponding photon wavelength.
Steps
Step 1: Identify the minimum photon energy required
The photon must carry at least the stated binding/threshold energy — treat that value as E directly. Convert it to electron-volts if it isn't already (e.g. 1 MeV=106 eV); electron-volts pair naturally with the shortcut in Step 3.
Step 2: Start from the photon energy–wavelength relation
E=λhc⇒λ=Ehc
Step 3: Use the hc≈1240 eV⋅nm shortcut
For any photon energy expressed in eV, the wavelength in nanometres is simply
λ(nm)=E(eV)1240
This avoids carrying h and c separately through the algebra and is accurate enough for exam purposes.
Step 4: Apply to this problem and sanity-check the order of magnitude
Divide 1240 by the energy in eV, watching the powers of ten carefully — a binding energy in the MeV range (nuclear scale) should give a wavelength many orders of magnitude shorter than a typical atomic-scale binding energy (eV range, giving hundreds of nm). If your answer doesn't fall in the expected range for the physical scale of the problem, re-check the unit conversion in Step 1 rather than the formula.
Showing the 12 most recent of 30 on this concept.
- CBSE 2026Set ANNUAL1 markQ.If the wavelength of a photon is halved, then its frequency will become ______.
›Reveal solutionSolution
Since nu = c/lambda for a photon, halving lambda directly doubles nu (c is a universal constant).
A photon's frequency and wavelength are related by nu = c/lambda, where c (speed of light) is fixed. If lambda is halved (lambda -> lambda/2), then nu = c/(lambda/2) = 2(c/lambda), i.e. the frequency becomes twice its original value.
✓Final answerdoubled (2x the original frequency).
- CBSE 2026Set ANNUAL1 markMCQQ.The mass of a photon is:(a) h/v(b) hc/λ(c) h/λ(d) hν/c²
›Reveal solutionSolution
A photon's energy is E=hν; equating this to E=mc2 gives its (relativistic/effective) mass m=hν/c2.
A photon has zero rest mass but carries energy E=hν (Planck's relation) and momentum p=h/λ=E/c. Using Einstein's mass-energy equivalence E=mc2 for the energy it carries while in motion, its effective mass is m=c2E=c2hν. The other options are quantities in disguise: hc/λ=hν is the photon's energy, not its mass, and h/λ is its momentum (p=h/λ), not its mass.
✓Final answer(d) hν/c2
- CBSE 2026Set ANNUAL1 markQ.Fill in the blank: The rest mass of photon is ______.
›Reveal solutionSolution
A photon's rest mass is zero.
A photon is a quantum of electromagnetic radiation that always travels at the speed of light c in vacuum. According to relativity, any particle moving at speed c must have zero rest mass; otherwise its energy would be infinite. A photon does have energy (E = hν) and momentum (p = hν/c), but its rest mass (mass measured when at rest) is zero.
✓Final answerzero.
- CBSE 2026Set ANNUAL1 markQ.Fill in the blank: The value of Planck's constant is ______.
›Reveal solutionSolution
Planck's constant h ≈ 6.63 × 10⁻³⁴ J·s.
Planck's constant h relates the energy of a photon to its frequency by E = hν. Its accepted value is
h = 6.63 × 10⁻³⁴ joule-second (J·s).
It is one of the fundamental constants of nature and appears throughout quantum physics.
✓Final answer6.63 × 10⁻³⁴ J·s.
- CBSE 2025Set 55/4/11 markMCQQ.A beam of red light and a beam of blue light have equal intensities. Which of the following statements is true? (A) The blue beam has more number of photons than the red beam. (B) The red beam has more number of photons than the blue beam. (C) Wavelength of red light is lesser than the wavelength of blue light. (D) The blue light beam has lesser energy per photon than that in the red light beam.
›Reveal solutionSolution
Since blue photons carry more energy than red photons, equal-intensity beams require more red photons to match the same total power. The red beam has more photons.
The key to this problem lies in understanding what "intensity" means and how photon energy depends on wavelength.
Intensity measures the power (energy per unit time) delivered per unit area. When two beams have equal intensities, they carry the same total energy per second through the same cross-sectional area, regardless of color.
Each photon carries energy E=hν=λhc, where h is Planck's constant, c is the speed of light, and λ is the wavelength. Blue light has a shorter wavelength than red light (λblue<λred), which means blue photons are individually more energetic than red photons.
If the total power delivered by both beams is the same, but blue photons pack more energy each, then fewer blue photons are needed to deliver that power. Conversely, more red photons are required to compensate for their lower individual energy.
Let me work through this quantitatively:
- Express intensity in terms of photon count. If n photons pass through area A in time t, the intensity is:
I=A⋅tTotal energy=A⋅tn⋅Ephoton=A⋅tn⋅hc/λ
- Set up the equal-intensity condition. For red and blue beams with equal intensities:
Ired=Iblue
A⋅tnred⋅hc/λred=A⋅tnblue⋅hc/λblue
- Simplify to find the photon ratio:
nred⋅λred1=nblue⋅λblue1
nbluenred=λblueλred
- Apply the wavelength relationship. Since red light has a longer wavelength than blue light (λred>λblue):
nbluenred>1⟹nred>nblue
Now let's check each option:
- (A) Claims blue has more photons — false, we just showed the opposite.
- (B) Claims red has more photons — true, matches our derivation.
- (C) Claims red wavelength is less than blue — false, red has longer wavelength.
- (D) Claims blue photons have less energy — false, Eblue=hc/λblue>hc/λred=Ered.
TipA quick mnemonic: "Lower energy photons need higher numbers" — to match the same total power, the beam with less energetic photons must have more of them.
✓Final answerThe correct option is (B): the red beam has more photons than the blue beam.
- CBSE 2025Set 55/5/11 markMCQQ.Which of the following electromagnetic waves has photons of the largest momentum? (A) X-rays (B) AM radio waves (C) Microwaves (D) TV waves
›Reveal solutionSolution
Photon momentum is p=λh, so the wave with the shortest wavelength has the largest momentum. Among the options, X-rays have the shortest wavelength, hence the largest photon momentum.
Concept & Intuition
The momentum of a photon is not like the momentum of a massive particle. For a photon, momentum is purely a wave property, given by the de Broglie relation:
p=λh
where h is Planck’s constant and λ is the wavelength. This means: shorter wavelength → larger momentum. There is no dependence on amplitude or intensity — only wavelength matters.
So the question reduces to: which of these electromagnetic waves has the shortest wavelength? Let’s recall the electromagnetic spectrum order from longest to shortest wavelength:
- Radio waves (including AM and TV) — longest wavelengths (metres to kilometres)
- Microwaves — centimetres to millimetres
- Infrared — micrometres
- Visible light — hundreds of nanometres
- Ultraviolet — tens of nanometres
- X-rays — picometres to nanometres
- Gamma rays — sub-picometre
Watch outA common mistake is to think that higher frequency means higher energy (true), but then incorrectly assume that momentum depends on something else like the wave’s “penetrating power” or “ionising ability”. Stick to p=h/λ — it’s the only formula that matters here.
Step-by-step solution
-
Write the momentum formula
For any photon, p=λh. Since h is constant, p∝λ1.
-
Identify the wavelengths of each option
- AM radio waves: wavelength ≈100 m to 1000 m (longest)
- TV waves: wavelength ≈0.1 m to 10 m (still radio band)
- Microwaves: wavelength ≈1 mm to 30 cm
- X-rays: wavelength ≈0.01 nm to 10 nm (shortest among these)
-
Compare
Since p∝1/λ, the smallest λ gives the largest p. X-rays have the smallest wavelength by many orders of magnitude.
-
Conclude
X-ray photons carry the largest momentum.
TipYou don’t need to memorise exact numbers — just remember the order of the EM spectrum from longest to shortest wavelength: Radio → Microwave → Infrared → Visible → UV → X-ray → Gamma. The one furthest to the right among the options wins.
✓Final answerThe correct option is (A) X-rays.
- CBSE 2025Set D1 markMCQQ.What is the energy of a photon with a wavelength of 500 nm? ( Use c = 3 × 10^8 m/s and h = 6.626 × 10^-34 Js ) (A) 4 × 10^-19 J (B) 2.5 × 10^-19 J (C) 1.2 × 10^-18 J (D) 6.6 × 10^-19 J
›Reveal solutionSolution
Photon energy E = hc/λ ≈ 4 × 10⁻¹⁹ J for λ = 500 nm.
The energy of a photon is
E=λhc
Substitute h = 6.626×10⁻³⁴ J·s, c = 3×10⁸ m/s, λ = 500 nm = 500×10⁻⁹ m:
E=500×10−9(6.626×10−34)(3×108)
E=5×10−71.9878×10−25=3.98×10−19 J
This rounds to 4 × 10⁻¹⁹ J.
✓Final answer(A) 4 × 10⁻¹⁹ J.
- CBSE 2025Set A1 markQ.Match Column 'A' item 'Frequency of light' with the correct option from Column 'B' and write the correct pair. Column 'B' options:(i) Minimum energy to emit electrons from the surface(ii) Minimum frequency to emit electrons from the surface(iii) Frequency of photon(iv) Number of photons(v) Moving particle(vi) Photon(vii) Einstein.
›Reveal solutionSolution
Frequency of light corresponds to option (iii): frequency of photon.
In the photon (particle) picture of light proposed by Einstein, a beam of light of frequency ν is regarded as a stream of photons, each carrying energy E = hν — so the 'frequency of light' (a wave concept) and the 'frequency of the photon' (used to compute each photon's quantum of energy) are simply the same physical quantity viewed from the two complementary (wave/particle) descriptions of light. Hence 'Frequency of light' matches '(iii) Frequency of photon'.
✓Final answerFrequency of light → (iii) Frequency of photon.
- CBSE 2025Set ANNUAL1 markQ.Electron volt (eV) is the unit of ................. (fill in the blank)
›Reveal solutionSolution
The electron volt is a convenient small unit of energy, widely used in atomic and nuclear physics.
One electron volt is defined as the kinetic energy gained by an electron when it is accelerated through a potential difference of 1 volt:
1 eV=1.6×10−19 J
Because atomic, photon, and nuclear energies are typically tiny fractions of a joule, the eV (and its multiples keV, MeV) is the standard convenient energy unit in this domain.
✓Final answerEnergy.
- CBSE 2025Set ANNUAL1 markQ.A blue lamp mainly emits light of wavelength 4500A∘. The lamp is rated at 150 W and 8% of energy is emitted as visible light. How many photons are emitted by lamp per second?
›Reveal solutionSolution
Visible-light power = 8% of 150 W; divide by the energy of one photon at 4500 Å.
Power emitted as visible light =8% of 150W =0.08×150=12W.
Energy of one photon at λ=4500A˚=4.5×10−7m:
E=λhc=4.5×10−76.63×10−34×3×108≈4.42×10−19 J
Number of photons emitted per second:
n=EP=4.42×10−1912≈2.71×1019 photons/s
✓Final answerAbout 2.71×1019 photons are emitted per second.
- CBSE 2025Set ANNUAL1 markMCQQ.The momentum of a photon of energy h.nu is(i) h.nu(ii) h.nu/c(iii) h.nu.c(iv) h/nu
›Reveal solutionSolution
Photon momentum p = E/c = h(nu)/c.
A photon of frequency ν carries energy E=hν. Being a massless quantum that moves at the speed of light, its momentum is p=E/c. Therefore p=chν (equivalently p=h/λ since c=νλ).
✓Final answer(ii) h.nu/c.
- CBSE 2024Set ANNUAL1 markMCQQ.The momentum (p) of photon is -(a) h/λ(b) λ/h(c) hC/λ(d) hλ
›Reveal solutionSolution
A photon's momentum follows from combining its energy E = hc/λ with the relativistic relation E = pc for a massless particle.
A photon of frequency ν has energy E=hν=λhc (since c=νλ).
A photon is massless and travels at speed c, so by the relativistic energy-momentum relation for a massless particle, E=pc. Equating the two expressions for E:
pc=λhc⟹p=λh
✓Final answer(a) h/λ.
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