Modern Physics — Photoelectric Effect, Atoms, Nuclei and Semiconductors
Modern physics is the highest-yield block in the physics syllabus per hour of study. The formulae are few, the calculations are short, and the questions are largely conceptual — which means a candidate who understands why each result holds can answer almost all of them, while one who memorised will be caught by the graph and the "what if" variants. The organising idea is quantisation: energy, angular momentum and charge all come in discrete lumps rather than continuously, and every surprising result in the chapter is a consequence of that.
The Bohr Atom and Spectra
Bohr quantised angular momentum as mvr = nh/2π, and for hydrogen that gives E_n = −13.6/n² eV and r_n = 0.53 n² Å. A transition from n₂ to n₁ emits a photon of energy E₂ − E₁, which is why atomic spectra are lines rather than a continuum.
| Series | Transitions to | Region |
|---|---|---|
| Lyman | n = 1 | Ultraviolet |
| Balmer | n = 2 | Visible — the only one we can see |
| Paschen | n = 3 | Infrared |
| Brackett / Pfund | n = 4 / n = 5 | Far infrared |
🧠 Reading the energy ladder, and the two sign facts
E_n = −13.6/n² eV gives −13.6 eV for the ground state, −3.4 for n = 2, −1.51 for n = 3, and 0 at n = ∞. Two consequences are asked constantly. The ionisation energy of hydrogen is 13.6 eV — the amount needed to lift an electron from n = 1 to infinity — and it is positive because you must supply it. And the levels crowd together as n rises, so successive spectral lines in a series get closer and converge to a limit. Note also that for a hydrogen-like ion of atomic number Z, E_n = −13.6Z²/n², so He⁺ has four times hydrogen's ionisation energy — a favourite one-step extension.Nuclei — Binding Energy, Decay and Fission
The mass of a nucleus is less than the sum of its constituent nucleons, and that missing mass is the binding energy through E = Δm c², with 1 u ≡ 931.5 MeV. What matters for stability is not the total binding energy but the binding energy per nucleon, which peaks near ⁵⁶Fe at about 8.8 MeV.
🎯 Why fission and fusion both release energy
This looks contradictory until you look at the binding-energy-per-nucleon curve. It rises steeply from hydrogen, peaks around iron, and then falls slowly towards uranium. Energy is released whenever a reaction moves nucleons towards the peak. So very light nuclei release energy by joining together — fusion, which powers the Sun — while very heavy nuclei release energy by splitting — fission, which powers reactors. Iron is the most stable nucleus and can do neither, which is also why stellar cores stop fusing at iron and then collapse. Per nucleon, fusion releases considerably more than fission, which is why it remains the more attractive prospect.| Emission | What it is | Change in Z and A |
|---|---|---|
| Alpha | A helium nucleus, ⁴₂He | Z − 2, A − 4 |
| Beta-minus | An electron, from a neutron turning into a proton | Z + 1, A unchanged |
| Beta-plus | A positron, from a proton turning into a neutron | Z − 1, A unchanged |
| Gamma | A high-energy photon | Neither changes — only energy is lost |
🧠 Half-life arithmetic, and why decay is exponential
The law is N = N₀e−λt with half-life T = 0.693/λ and mean life τ = 1/λ, so τ is always the longer of the two. For exam arithmetic use the powers of two rather than logarithms: after n half-lives the fraction remaining is (½)ⁿ, so three half-lives leave ⅛ and four leave ¹⁄₁₆. The deeper point is that decay is a probabilistic process — each nucleus has a fixed chance of decaying per second regardless of how long it has already existed, so nuclei do not "age". That is why the half-life is completely unaffected by temperature, pressure or chemical state, which is exactly what makes radiometric dating trustworthy.Semiconductors and Diodes
| Class | Band gap | Resistance with rising temperature |
|---|---|---|
| Conductor | None — bands overlap | Increases |
| Semiconductor | Small — about 1.1 eV for Si, 0.7 eV for Ge | Decreases |
| Insulator | Large — 3 eV or more | Decreases, but stays enormous |
🎯 Why heating helps a semiconductor and hurts a metal
In a metal the number of free electrons is already fixed and enormous; heating only makes the lattice vibrate more, which scatters those electrons more often, so resistance rises. In a semiconductor the number of carriers is tiny at room temperature, and heating lifts many more electrons across the small band gap, so the carrier count rises far faster than the extra scattering hurts, and resistance falls — a negative temperature coefficient. That single contrast explains why a thermistor makes a good temperature sensor and why silicon devices need heat sinks: as they warm they conduct better, drawing more current, warming further, which is a runaway unless controlled.🧠 Doping, the p–n junction and how to read a diode question
Adding a pentavalent impurity (P, As, Sb) to silicon donates spare electrons and gives an n-type semiconductor; a trivalent one (B, Al, In) creates holes and gives p-type. Note that both remain electrically neutral — doping adds carriers, not charge, and thinking otherwise is a common error. Join them and diffusion creates a depletion layer with a barrier potential of about 0.7 V for silicon and 0.3 V for germanium. In forward bias (p to positive) the barrier narrows and current flows freely; in reverse bias it widens and only a tiny leakage current flows. So a diode conducts one way, which is exactly what makes rectification possible: a half-wave rectifier uses one diode and passes one half of each cycle, a full-wave bridge uses four and passes both.Key takeaways
- hν = φ + KE_max; intensity changes the NUMBER of electrons, frequency changes their maximum KE.
- Below the threshold frequency nothing is emitted at any intensity, and emission is instantaneous.
- Stopping potential against frequency is a straight line of slope h/e — the SAME slope for every metal.
- λ = h/p for matter too; an electron at 100 V has λ ≈ 1.2 Å, comparable to atomic spacing.
- E_n = −13.6/n² eV; ionisation energy 13.6 eV; only the Balmer series is visible.
- Binding energy per nucleon peaks at iron, so light nuclei fuse and heavy nuclei fission.
- Alpha: Z−2, A−4. Beta-minus: Z+1, A same. Gamma: neither changes. After n half-lives, (½)ⁿ remains.
- A semiconductor's resistance FALLS with temperature; a metal's rises. Doped semiconductors stay neutral.
Practice questions (8)
Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.
In the photoelectric effect, doubling the intensity of the incident light at fixed frequency:
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Answer: B — Doubles the number of electrons but leaves their maximum KE unchanged
Intensity is the number of photons per second, so doubling it doubles the electrons ejected. But each electron interacts with a single photon of energy hν, so its maximum kinetic energy — and the stopping potential — depends only on the frequency. This separation of "how many" from "how energetic" is what the wave theory could not explain.A graph of stopping potential against frequency is plotted for two different metals. The two lines:
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Answer: B — Have the same slope h/e but different intercepts
From eV₀ = hν − φ, the slope is h/e and is universal, while the intercept −φ/e depends on the metal. That the slope is identical for every metal is precisely what established h as a fundamental constant and let this experiment measure it directly.Which series of the hydrogen spectrum lies in the visible region?
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Answer: B — Balmer
Balmer transitions end at n = 2 and fall in the visible band. Lyman ends at n = 1 and is ultraviolet, since transitions to the ground state involve the largest energy gaps; Paschen and Brackett end at n = 3 and n = 4 and are infrared.Both fission of heavy nuclei and fusion of light nuclei release energy because:
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Answer: B — Both move nucleons towards the peak of the binding-energy-per-nucleon curve
The curve peaks near iron at about 8.8 MeV per nucleon, so light nuclei gain binding energy by joining and heavy nuclei gain it by splitting — both approach the peak. Iron itself can do neither, which is why stellar cores stop fusing there. Nucleon number is conserved; it is mass that converts to energy via E = Δmc².When a nucleus emits a beta-minus particle, its atomic number Z and mass number A change as:
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Answer: B — Z increases by 1, A stays the same
In beta-minus decay a neutron converts into a proton and emits an electron, so the proton count rises by one while the total nucleon count is unchanged. Option 1 describes alpha decay, option 3 beta-plus decay, and option 4 gamma emission, which carries away energy only.After three half-lives, the fraction of a radioactive sample remaining undecayed is:
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Answer: C — 1/8
Each half-life halves the sample, so three leave (½)³ = 1/8. Decay is exponential rather than linear, which is why 1/3 is wrong — and it never reaches exactly zero, because each nucleus decays with a fixed probability per second regardless of how long it has already survived.As the temperature of a pure semiconductor rises, its electrical resistance:
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Answer: B — Decreases, because more carriers are freed across the band gap
A semiconductor has few carriers at room temperature, and heating lifts many more electrons across its small band gap, so the carrier count rises much faster than the extra lattice scattering hurts. A metal already has a fixed, enormous carrier population, so for it only the scattering matters and resistance rises.An n-type semiconductor is formed by doping silicon with a pentavalent impurity. The resulting material is:
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Answer: B — Electrically neutral overall
Each donor atom contributes a spare electron but also an extra proton in its own nucleus, so the material stays neutral — "n-type" refers to the majority carrier being negative, not to a net charge. Assuming an n-type crystal is negatively charged is a standard misconception, and the same applies in mirror image to p-type.