Modern Physics

Modern physics carries about 2 questions (our estimate), but the syllabus is long: electrons, the photoelectric effect, Bohr, de Broglie, X-rays, semiconductors and gates, radioactivity and nuclear energy, plus a list of "recent trends" (particle physics, the universe, seismology, telecommunication, environment, new materials). The numerical questions repeat a few formulae; the recent-trends items are definitions you can score by reading once.

6.1 Electrons

Cathode rays (a discharge tube at low pressure) are streams of electrons: deflected by electric and magnetic fields, negatively charged, travel in straight lines, produce X-rays on striking a target. J.J. Thomson: with crossed fields the speed is v=\frac{E}{B}; the deflection with only B gives the specific charge \frac{e}{m}=1.76\times10^{11} C kg⁻¹. Millikan's oil-drop: drops balanced between plates, qE=mg; every charge is an integral multiple of e=1.6\times10^{-19} C (charge is quantised). Electron mass 9.1\times10^{-31} kg. An electron accelerated through V volts gains eV joules (1\ \text{eV}=1.6\times10^{-19} J) and speed \sqrt{\frac{2eV}{m}}.

6.2 Photons and quantization of energy

Photoelectric effect. Light of frequency above the threshold f_0 ejects electrons instantly; K_{max} depends on frequency, not intensity; the photo-current depends on intensity.

h=6.63\times10^{-34} J s; photon energy E=\frac{hc}{\lambda}=\frac{1240}{\lambda(\text{nm})} eV; photon momentum \frac{h}{\lambda}. The wave theory cannot explain the instantaneous emission or the threshold. A 100 W sodium lamp (589 nm) emits about 3\times10^{20} photons per second.

Bohr's theory (hydrogen): angular momentum mvr=\frac{nh}{2\pi}; E_n=-\dfrac{13.6}{n^2} eV; r_n=0.53n^2 Å; v_n=\frac{2.2\times10^6}{n} m s⁻¹. Emission when an electron falls: h f=E_{n_2}-E_{n_1}, \frac{1}{\lambda}=R\Big(\frac{1}{n_1^2}-\frac{1}{n_2^2}\Big), R=1.097\times10^7 m⁻¹.

Ionisation energy of hydrogen is 13.6 eV; the first excitation energy 10.2 eV.

de Broglie: \lambda=\frac{h}{p}=\frac{h}{mv}; electron through V volts: \lambda=\frac{1.227}{\sqrt{V}} nm (V=150 V gives 0.1 nm). Heisenberg uncertainty: \Delta x\Delta p\ge\frac{\hbar}{2}=\frac{h}{4\pi}. X-rays: produced when fast electrons strike a metal target; the continuous spectrum has a minimum wavelength \lambda_{min}=\frac{hc}{eV}; the characteristic lines depend on the target (Moseley). Bragg's law: 2d\sin\theta=n\lambda. X-rays are not deflected by fields, penetrate matter, are used in medicine and crystallography. Laser: stimulated emission, population inversion, optical cavity; light is coherent, monochromatic, directional and intense (surgery, cutting, communication, barcode readers).

6.3 Solids and semiconductor devices

Band theory: conductors (bands overlap), insulators (gap >3 eV), semiconductors (gap about 1 eV: Si 1.1, Ge 0.7). Intrinsic semiconductors have equal electron and hole numbers; their resistance falls with temperature. Extrinsic: n-type (donor, pentavalent: P, As, Sb; electrons majority), p-type (acceptor, trivalent: B, Al, In; holes majority); n_en_h=n_i^2.

p-n junction: diffusion creates a depletion layer and a barrier potential (0.3 V Ge, 0.7 V Si). Forward bias (p to +): the layer narrows and current flows; reverse bias: the layer widens, a tiny leakage current flows. Rectification: half-wave (one diode, ripple frequency = input), full-wave (two diodes or a bridge, ripple 2f); a capacitor smooths the output. Zener diode: in reverse breakdown it holds a constant voltage (a regulator); LED and photodiode are other junction devices.

Transistor (npn / pnp; emitter, base, collector): I_E=I_B+I_C; \alpha=\frac{I_C}{I_E}, \beta=\frac{I_C}{I_B}, \beta=\frac{\alpha}{1-\alpha} (\alpha=0.98: \beta=49); amplifier in common-emitter mode (phase reversal 180^\circ), switch in cut-off and saturation. Logic gates:

De Morgan: \overline{A\cdot B}=\bar A+\bar B, \overline{A+B}=\bar A\cdot\bar B.

6.4 Radioactivity and nuclear reaction

Nucleus: A=Z+N; radius R=R_0A^{1/3}, R_0=1.2 fm; so density is constant, about 2.3\times10^{17} kg m⁻³, independent of A. Isotopes share Z; isobars share A; isotones share N. Atomic mass unit: 1 u =1.66\times10^{-27} kg =931.5 MeV/c^2.

Mass–energy: E=mc^2; mass defect \Delta m=Zm_p+Nm_n-m_{nucleus}; binding energy =\Delta m\times931.5 MeV; BE per nucleon peaks near ^{56}Fe (8.8 MeV), so fission of heavy and fusion of light nuclei both release energy. Conversion of 1 g of matter gives 9\times10^{13} J.

Decay law: N=N_0e^{-\lambda t}=N_0\big(\frac{1}{2}\big)^{t/T_{1/2}}; T_{1/2}=\frac{\ln2}{\lambda}=\frac{0.693}{\lambda}; mean life \tau=\frac{1}{\lambda}=1.44T_{1/2}; activity A=\lambda N (becquerel = 1 decay s⁻¹; curie =3.7\times10^{10} Bq). After n half-lives \big(\frac{1}{2}\big)^n remains. \alpha decay: Z-2, A-4; \beta^-: Z+1, an antineutrino emitted; \gamma: no change in Z,A.

Carbon dating: living matter has a constant ^{14}C/^{12}C ratio; after death ^{14}C (T_{1/2}=5730 y) decays; a sample with 25 % of the original is 11 460 years old. Fission: ^{235}U + n → fragments + 2–3 neutrons + about 200 MeV, a chain reaction controlled by moderator (graphite, heavy water) and control rods (Cd, B). Fusion: light nuclei combine at about 10^7 K (Sun: proton–proton cycle). Health hazards: ionising radiation damages DNA; dose in sievert; shielding, distance and time reduce exposure; radioactive waste handling.

6.5 Recent trends in physics

• Particle physics: every particle has an antiparticle (positron–electron annihilation \to2\gamma). Quarks (up, down, strange, charm, top, bottom; fractional charge \pm\frac{1}{3},\pm\frac{2}{3}) build hadrons: baryons (three quarks: proton uud, neutron udd) and mesons (quark–antiquark, e.g. pion). Leptons (electron, muon, tau and their neutrinos) are fundamental. The Higgs boson (found at the LHC, CERN, 2012; mass about 125 GeV) is the quantum of the field that gives mass.
• Universe: Big Bang about 13.8 billion years ago; Hubble's law v=H_0d (galaxies recede; expansion); the cosmic microwave background; dark matter (galaxy rotation curves) and dark energy; gravitational waves (ripples of space-time from merging black holes, first detected by LIGO in 2015); black hole: escape speed exceeds c, event horizon r_s=\frac{2GM}{c^2}.
• Seismology: P-waves (longitudinal, fastest, through solids and liquids), S-waves (transverse, solids only); surface (L) waves cause the greatest damage; the Richter / magnitude scale is logarithmic.
• Telecommunication: radio (AM, FM, modulation), TV, mobile (cells), GPS (24+ satellites, timing signals; relativistic corrections), remote sensing (satellite imaging of land, forests, floods).
• Environment: energy crisis, greenhouse effect and global warming, ozone layer depletion by CFCs (UV-B), pollution.
• New materials: nanotechnology (1–100 nm; properties change with size), superconductors (zero resistance below T_c, Meissner effect — expel magnetic field; used in MRI and maglev), distinguished from a perfect conductor (which does not expel fields).

Formula and fact sheet

Practice questions

Set A — Recall (10 questions)

1. The specific charge of the electron isa1.76\times10^{11} C kg⁻¹b9.1\times10^{-31} C kg⁻¹c1.6\times10^{-19} C kg⁻¹d1.76\times10^{-11} C kg⁻¹
2. The energy of a photon of wavelength 620 nm isa1 eVb2 eVc4 eVd0.5 eV
3. Photons of energy 5 eV fall on a metal of work function 2 eV. The stopping potential isa2 Vb5 Vc3 Vd7 V
4. The energy of the hydrogen atom in the n=2 level isa-13.6 eVb-6.8 eVc-1.51 eVd-3.4 eV
5. The Balmer series of hydrogen lies in theavisible regionbultravioletcinfrareddX-ray region
6. The de Broglie wavelength of an electron accelerated through 150 V is abouta1 nmb0.1 nmc0.01 nmd10 nm
7. After three half-lives the fraction of a radioactive sample that remains isa1/3b1/6c1/8d1/9
8. An energy equivalent to 1 atomic mass unit isa9.3 MeVb93 MeVc9315 MeVd931.5 MeV
9. In forward bias the depletion region of a p-n junctionanarrowsbwidenscis unchangeddis permanent
10. Which gate is known as a universal gate?aANDbORcNANDdXOR

Set B — Understanding (6 questions)

1. A metal has threshold wavelength 500 nm. Light of wavelength 400 nm falls on it. The maximum kinetic energy of the photoelectrons is nearlya0.62 eVb1.24 eVc3.1 eVd2.48 eV
2. The wavelength of the H\alpha line (the n=3\to2 transition) of hydrogen is nearlya486 nmb656 nmc434 nmd122 nm
3. A sample of 80 g of a radioactive element has a half-life of 10 days. After 30 days the mass remaining isa40 gb20 gc10 gd5 g
4. A wood sample has 25 % of the ^{14}C of a living sample (T_{1/2}=5730 y). Its age is abouta5730 yb2865 yc17190 yd11460 y
5. X-rays reflect from planes spaced 0.2 nm at a glancing angle of 30^\circ in first order. The wavelength isa0.2 nmb0.1 nmc0.4 nmd0.346 nm
6. The energy released by the complete conversion of 1 g of matter isa9\times10^{16} Jb9\times10^{13} Jc9\times10^{10} Jd3\times10^{8} J

Set C — Application (4 questions)

1. A 100 W lamp emits light of wavelength 589 nm, all of its power. The number of photons per second is abouta3\times10^{20}b3\times10^{18}c3\times10^{22}d3\times10^{19}
2. A nucleus has a mass defect of 0.03 u. Its binding energy is nearlya2.8 MeVb28 MeVc280 MeVd0.28 MeV
3. A transistor has \alpha=0.98. Its current gain \beta isa98b4.9c49d0.02
4. An electron is localised to within 1 Å. The minimum uncertainty in its momentum (\hbar=1.05\times10^{-34} J s) is abouta5.3\times10^{-24}b6.6\times10^{-24}c1.1\times10^{-34}d5.3\times10^{-25} kg m s⁻¹