JEE Physics Modern Physics Notes Set 01

Photoelectric Effect

1. Light of frequency above the threshold frequency \( \nu_0 \) ejects electrons from a metal almost instantly. Below \( \nu_0 \) no electrons are emitted whatever the intensity.
2. Einstein's equation: \( h\nu = \phi + K_{max} \), where \( \phi = h\nu_0 \) is the work function. The stopping potential is \( eV_0 = K_{max} \).
3. The number of photoelectrons per second (saturation current) is proportional to intensity. \( K_{max} \) and \( V_0 \) depend on frequency only. The graph of \( V_0 \) against ν is a straight line of slope h/e.
4. Photon: energy \( E = h\nu = hc/\lambda \), momentum \( p = h/\lambda = E/c \), zero rest mass. In convenient units \( E\,(\text{eV}) = 12400/\lambda\,(\text{Å}) \).

Illustration 1. Light of wavelength 4000 Å falls on a metal of work function 2.2 eV. Find the stopping potential.
Answer:
Photon energy \( = 12400/4000 = 3.1 \) eV.
\( eV_0 = 3.1 - 2.2 = 0.9 \) eV, so \( V_0 = 0.9 \) V.

de Broglie Waves

1. Every moving particle has a wavelength \( \lambda = h/p = h/mv \).
2. For a particle of kinetic energy K: \( \lambda = h/\sqrt{2mK} \). For an electron accelerated through V volts: \( \lambda = 12.27/\sqrt V \) Å. For a particle of charge q: \( \lambda = h/\sqrt{2mqV} \).
3. For a gas atom at temperature T: \( \lambda = h/\sqrt{3mkT} \).
4. The Davisson–Germer experiment confirmed the wave nature of electrons by diffraction from a nickel crystal. The electron speed rises with the accelerating potential.

Bohr Model of the Hydrogen Atom

1. For hydrogen-like ions of atomic number Z: radius \( r_n = 0.529\,n^2/Z \) Å, speed \( v_n = 2.18\times10^6\,Z/n \) m/s, energy \( E_n = -13.6\,Z^2/n^2 \) eV.
2. The angular momentum is \( mvr = nh/2\pi \), and the circumference of an orbit is \( n\lambda \). Kinetic energy = −(total energy).
3. Spectral lines: \( \dfrac1\lambda = RZ^2\left(\dfrac1{n_1^2}-\dfrac1{n_2^2}\right) \), with \( R = 1.097\times10^7 \) m⁻¹. Lyman series ends on n = 1 (ultraviolet), Balmer on n = 2 (visible), Paschen on n = 3 (infrared).
4. An atom excited to level n can emit up to \( n(n-1)/2 \) different lines.
5. Recoil speed after a photon of wavelength λ is emitted by an atom of mass m: \( v = h/m\lambda \).

Illustration 2. Find the ratio of the longest wavelength of the Lyman series to the longest wavelength of the Balmer series.
Answer:
Lyman: \( 1/\lambda = R(1-1/4) = 3R/4 \). Balmer: \( 1/\lambda = R(1/4-1/9) = 5R/36 \).
Ratio \( = \dfrac{4/3R}{36/5R} = \dfrac5{27} \).

X-rays

1. X-rays come from the sudden stopping of fast electrons on a metal target. The continuous spectrum has a minimum wavelength \( \lambda_{min} = hc/eV = 12400/V\,(\text{Å}) \), which decreases as the tube voltage increases.
2. Characteristic X-rays are produced by transitions to inner shells and are a property of the target element. Moseley's law: \( \sqrt\nu\propto(Z-1) \), so \( \lambda\propto1/(Z-1)^2 \).
3. Harder X-rays have higher frequency and higher photon energy.

Nucleus

1. Nuclear radius \( R = R_0A^{1/3} \) with \( R_0 = 1.2 \) fm. Nuclear density is about \( 10^{17} \) kg/m³ and is the same for all nuclei.
2. Mass defect \( \Delta m = ZM_p + NM_n - M \) and binding energy \( = \Delta m\times931.5 \) MeV. The binding energy per nucleon peaks near A = 56. Fission of heavy nuclei and fusion of light nuclei both release energy.
3. Nuclear forces between pp, pn and nn pairs are equal at the same separation, are short range and are independent of charge.
4. α decay lowers A by 4 and Z by 2. β⁻ decay raises Z by 1. β⁺ decay lowers Z by 1. γ decay changes neither A nor Z.
5. Fission: \( \ce{^{235}_{92}U + ^1_0n -> ^{144}_{56}Ba + ^{89}_{36}Kr + 3 ^1_0n} \) + energy. Heavy water, graphite and ordinary water are used as moderators, and cadmium rods absorb neutrons to control the reaction.

Radioactivity

1. Decay law: \( N = N_0e^{-\lambda t} = N_0(1/2)^{t/T} \). Half-life \( T = 0.693/\lambda \) and mean life \( \tau = 1/\lambda = 1.44T \).
2. Activity \( A = \lambda N \) in becquerel or decays per second; \( A = A_0e^{-\lambda t} \). The α, β and γ rays deflect in an electric field towards the negative plate, towards the positive plate and not at all.
3. The β spectrum is continuous, because a neutrino or antineutrino shares the energy. The mass of a decaying nucleus is converted to energy by \( E = \Delta mc^2 \). If an object of mass M at rest emits an α-particle, the Q-value divides inversely as the masses.
4. Radiocarbon dating compares the present ¹⁴C activity of a sample with that of a living organism.
5. The fraction left after n half-lives is \( 1/2^n \). Time for N₀ to fall to N: \( t = \tfrac1\lambda\ln(N_0/N) \).

Illustration 3. A radioactive sample has half-life 20 minutes. If there are initially 1200 nuclei, find the time for 900 of them to disintegrate.
Answer:
Remaining nuclei = 300 = 1200/4, i.e. two half-lives.
Time = 2 × 20 = 40 minutes.