Thermal Physics, Waves, Optics and Modern Physics for the Olympiad — NSEP and INPhO

Weightage: After mechanics and electromagnetism, these four areas share the remaining marks in NSEP and INPhO. They are shorter, with standard formulas, and they reward fast, clean calculation. Confirm the syllabus and the exact paper pattern with HBCSE.

1. Kinetic theory and thermodynamics

For an ideal gas $PV=nRT$, with mean translational energy $\frac{3}{2}kT$ per molecule and root-mean-square speed $v_{rms}=\sqrt{3kT/m}$. The internal energy is $U=\frac{f}{2}nRT$ with $f$ degrees of freedom: 3 for a monatomic gas and 5 for a diatomic gas at ordinary temperatures. Then $\gamma =C_p/C_v=1+2/f$.

The first law is $\Delta U=Q-W$. Standard processes:

Worked example. A monatomic gas ($\gamma =5/3$) is compressed adiabatically to half its volume. Then $T_2/T_1=(V_1/V_2{)}^{\gamma -1}=2^{2/3}=1.59$, so it heats by nearly 60 percent.

A Carnot engine has efficiency $1-T_C/T_H$, and entropy change for a reversible process is $\Delta S=\int dQ/T$.

Heat transfer. Conduction gives $\frac{dQ}{dt}=kA\frac{\Delta T}{L}$, and Newton's law of cooling gives an exponential approach to the surroundings. A black body radiates $P=\sigma A{T}^4$, and a body at temperature $T$ in surroundings at $T_0$ loses a net $\sigma A(T^4-T_0^4)$. Wien's law is ${\lambda }_{max}T=2.9\times 10^{-3}$ m K.

2. Waves and sound

The speed of a wave on a string is $v=\sqrt{T/\mu }$, and of sound in a gas $v=\sqrt{\gamma P/\rho }$. A string fixed at both ends of length $L$ has harmonics $f_n=\frac{nv}{2L}$. An air column closed at one end supports only odd harmonics, $f_n=\frac{nv}{4L}$ for odd $n$.

Worked example. A string of length $0.8$ m, tension $100$ N and linear density $0.01$ kg/m has $v=100$ m/s and fundamental frequency $f_1=\frac{100}{2\times 0.8}=62.5$ Hz.

Superposition of waves gives interference and beats at frequency $|f_1-f_2|$. The Doppler effect for a source moving at $v_s$ and an observer at $v_o$, both along the line joining them, is:

$f^{\prime }=f\frac{v+v_o}{v-v_s}$

with $v_o$ positive towards the source and $v_s$ positive towards the observer. Sound intensity level is $10{\log }_{10}(I/I_0)$ dB, so doubling intensity adds about 3 dB.

3. Geometrical and wave optics

For a thin lens, $\frac{1}{f}=\frac{1}{v}-\frac{1}{u}$ (sign convention applied) and $\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$. Two thin lenses in contact have power $P=P_1+P_2$. Use Snell's law $n_1\sin {\theta }_1=n_2\sin {\theta }_2$ and total internal reflection at $\sin {\theta }_c=n_2/n_1$.

Young's double slit. Bright fringes occur where the path difference is $d\sin \theta =m\lambda$, and the fringe width on a screen at distance $D$ is $\beta =\frac{\lambda D}{d}$.

Worked example. With $\lambda =600$ nm, $D=1$ m and $d=1$ mm, $\beta =\frac{600\times 10^{-9}\times 1}{10^{-3}}=0.6$ mm.

For a thin film in air, reflected light from the two surfaces differs by a phase change of $\pi$ at one reflection, so the condition for a bright fringe is $2nt\cos r=(m+\frac{1}{2})\lambda$. Single-slit diffraction has minima at $a\sin \theta =m\lambda$. Brewster's angle satisfies $\tan {\theta }_B=n$, where reflected light is fully polarised. The resolving limit of a circular aperture is about $1.22\lambda /D$.

4. Quantum and atomic physics

The photoelectric effect: $K_{max}=hf-\varphi$, and no electrons are emitted below the threshold frequency, whatever the intensity. It is convenient to use $hc=1240$ eV nm.

Worked example. Light of wavelength $250$ nm has $E=1240/250=4.96$ eV. For a metal with work function $2.0$ eV, $K_{max}=2.96$ eV, and the stopping potential is $2.96$ V.

The de Broglie wavelength is $\lambda =h/p$. The Bohr model for hydrogen gives energies $E_n=-13.6/n^2$ eV and radii $r_n=n^2{a}_0$ with $a_0=0.053$ nm.

Worked example. The transition $n=3\to 2$ emits $13.6\left(\frac{1}{4}-\frac{1}{9}\right)=1.89$ eV, a wavelength of $1240/1.89=656$ nm, the red Balmer line.

5. Nuclear physics

Radioactive decay obeys $N=N_0{e}^{-\lambda t}$, with half-life $t_{1/2}=\ln 2/\lambda$. After $n$ half-lives a fraction $2^{-n}$ remains, so a sample with an 8-day half-life retains one-eighth after 24 days. The activity is $A=\lambda N$. Binding energy follows from the mass defect through $E=\Delta m{c}^2$, with $1$ u $=931.5$ MeV$/c^2$.

6. Special relativity

The Lorentz factor is $\gamma =\frac{1}{\sqrt{1-v^2/c^2}}$. Time dilation $\Delta t=\gamma \Delta t_0$, length contraction $L=L_0/\gamma$, and the relativistic energy $E=\gamma m{c}^2$, with $E^2=(pc{)}^2+(m{c}^2{)}^2$.

Worked example. At $v=0.6c$, $\gamma =1/\sqrt{1-0.36}=1.25$. A clock that records $1\mu$s in the moving frame is measured to take $1.25\mu$s in the lab frame.

Velocities add by $u^{\prime }=\frac{u+v}{1+uv/c^2}$, so nothing exceeds $c$.

7. Estimating and checking

For every result, check units and compare with a plausible value: a photon of visible light has an energy of about 2 to 3 eV, the speed of sound in air is about 340 m/s, and an atom's size is about $10^{-10}$ m. A result far outside such a range signals an error.

Common traps

• Using isothermal work for an adiabatic process.
• Using the wrong sign convention in the lens formula or Doppler formula.
• Forgetting the extra half-wavelength in a thin-film condition.
• Using temperature in Celsius in radiation or gas-law formulas.
• Treating the photoelectric threshold as depending on intensity.

Memory aids

• "Three, five, and one plus two over f": degrees of freedom and gamma.
• "1240 eV nm": photon energy and wavelength.
• "Two to the minus n": fraction left after n half-lives.

Summary

Thermal physics rests on the ideal gas, the first law and the four standard processes, with radiation and conduction for heat flow. Waves use $v=\sqrt{T/\mu }$, standing-wave harmonics and the Doppler formula.

Optics covers lenses, interference and diffraction, and modern physics uses the photoelectric equation, the Bohr model, decay law and the Lorentz factor.

Exam protocol

• Convert to kelvin and SI units first.
• State the sign convention before using a lens or Doppler formula.
• Use $hc=1240$ eV nm for photon problems.
• Check a result against a physical estimate.