What is the Bohr model of the atom?

Bohr's 1913 model says an electron in hydrogen can only orbit at certain fixed radii where its angular momentum is an integer multiple of h/2pi. While in an orbit it radiates no energy; it only emits or absorbs a photon when it jumps between orbits, with photon energy equal to the energy difference between them. This exactly predicts hydrogen's line spectrum.

What is the photoelectric effect?

Light shining on a metal surface can eject electrons, but only if each photon carries at least the metal's work function energy — below that threshold frequency, no electrons are emitted no matter how bright the light. Einstein explained this in 1905 with E=hf=phi+KE_max, which won him the Nobel Prize and confirmed light behaves as discrete photons.

What is a semiconductor band gap?

Electrons in a solid can only occupy certain energy bands. In conductors the valence and conduction bands overlap so electrons flow freely; in insulators a large gap (over about 5 eV) blocks conduction; in semiconductors a small gap (around 1 eV) lets a few electrons cross at room temperature, and doping with donor or acceptor impurities adds extra energy levels that dramatically increase conductivity.

What is the difference between n-type and p-type semiconductors?

n-type semiconductors are doped with donor atoms (like phosphorus in silicon) that contribute extra free electrons as the majority charge carrier. p-type semiconductors are doped with acceptor atoms (like boron) that create extra holes — missing electrons that behave as positive charge carriers.

Forces / Modern physics

Modern physics: three ideas explained

Three class 12 ideas that don't need a full sandbox to click, just a clear animated picture: Bohr's quantised hydrogen orbits and the line spectrum they produce (1913), Einstein's photoelectric effect — proof that light comes in discrete photons (1905), and the energy-band picture that explains why conductors, insulators and semiconductors behave so differently.

a quantised jump emits one photon, E=hν

1913

Bohr's model of hydrogen

Niels Bohr — quantised angular momentum

Bohr proposed that an electron in hydrogen can only orbit at radii where its angular momentum is a whole-number multiple of h/2π — mvr = nh/2π, n = 1, 2, 3… While in one of these allowed orbits the electron radiates no energy, so the atom is stable. It only emits or absorbs energy when it jumps between orbits, as a single photon whose energy equals the exact energy gap between them.

The formulas that make it work

Orbit radius: r_n = n²a₀ (a₀ = 0.529 Å, the Bohr radius).
Energy of orbit n: E_n = −13.6/n² eV — always negative (electron is bound), least negative (closest to free) as n grows.
A jump from n_i to n_f emits a photon of energy E_i−E_f; grouping by which shell the electron lands on gives the named series — Lyman (lands on n=1, UV), Balmer (n=2, visible), Paschen (n=3, infrared).

ExplainsHydrogen's exact line spectrum — every observed wavelength matches E_i−E_f to high precision, the model's biggest triumph.

LimitOnly works cleanly for one-electron atoms (H, He⁺). It treats the electron as a ball on a fixed path, which the uncertainty principle rules out — the quantum model replaced it.

a photon above the threshold frequency ejects an electron instantly

1905 (Nobel Prize 1921)

The photoelectric effect

Albert Einstein — light as discrete photons

Shine light on a metal and it can knock electrons out — but only above a minimum threshold frequency, no matter how bright the light. A dim light above threshold ejects electrons instantly; an intense light below threshold ejects none at all. Classical wave theory (energy = brightness) cannot explain this. Einstein could, by treating light as a stream of photons, each carrying a fixed energy E = hf.

Einstein's equation

Each photon transfers all its energy to one electron: hf = φ + KE_max, where φ (the work function) is the minimum energy needed to free an electron from that metal.
Below the threshold frequency f₀=φ/h, no electron is ejected — extra brightness just means more (still too-weak) photons, not more energy per photon.
Above threshold, KE_max rises linearly with frequency, not brightness — brightness only increases how many electrons are ejected per second (the photocurrent).

ExplainsThe sharp frequency threshold and the instant response — both impossible for light-as-a-continuous-wave, both natural for light-as-photons.

LimitIt doesn't erase wave behaviour — diffraction and interference still need light-as-a-wave. Light needs both pictures (wave–particle duality).

bigger gap = harder for an electron to become a free carrier

~1928–1931

Semiconductor energy bands

Bloch & Wilson — band theory of solids

Electrons in a solid can't take just any energy — they fill discrete bands. The highest filled band is the valence band; the next one up, where electrons can move freely and carry current, is the conduction band. Whether a material conducts, insulates, or does something in between comes down entirely to the size of the gap between them.

Three materials, one picture

Conductor — the bands overlap (or the conduction band is already partly filled), so electrons flow with almost no push needed.
Insulator — a large gap (E_g > ~5 eV) that thermal energy at room temperature can't bridge, so essentially no free carriers exist.
Semiconductor — a small gap (E_g ≈ 1 eV, e.g. 1.1 eV for silicon) that a few electrons cross by thermal energy alone, and that doping exploits: donor impurities (like phosphorus in silicon) add easy-to-free electrons for an n-type semiconductor; acceptor impurities (like boron) create holes — missing electrons that act as positive carriers — for a p-type one.

ExplainsWhy silicon conducts a little (and much more when doped or heated), while diamond — same crystal structure, bigger gap — never does at room temperature.

LimitIt's a single-electron approximation; real conduction also depends on carrier mobility, temperature and crystal defects that the simple band picture leaves out.

	Conductor	Semiconductor	Insulator
Band gap E_g	~0 eV (overlap)	~0.7–1.5 eV	> ~5 eV
Example	Copper	Silicon, germanium	Diamond, glass
Conductivity rises with	— (falls with heat)	heat, light, doping	— (stays ~0)

Part of Forces — these are explainers, not interactive sandboxes; for hands-on labs see Electricity & magnetism, Optics and the rest of Forces. See Learn for the full class 12 modern physics syllabus this maps to.