(a) Explain classical theory of diamagnetism. Show that the susceptibility of diamagnetic substances is directly proportional to…

(a) Classical (Langevin) theory of diamagnetism

Theory: an atom with no permanent moment, placed in a field B, has each electron's orbit altered by Larmor's theorem. The electron's motion is the same as before except for an added precession about B with angular velocity ω_L = eB/2m. This is an induced circulating current whose moment opposes B, as Lenz's law requires. Induced moment: the extra current of one electron is I = −e(ω_L/2π), through the area πρ², where ρ is the distance from the field axis. So Δμ = −e(ω_L/2π)πρ² = −(e²B/4m)⟨ρ²⟩. For a spherically symmetric atom, ⟨ρ²⟩ = (2/3)⟨r²⟩, so Δμ = −(e²B/6m)⟨r²⟩ per electron. Susceptibility: with Z electrons and N atoms per unit volume, M = NZΔμ and B = μ₀H, so χ = M/H = −μ₀NZe²⟨r²⟩/(6m). For atoms of similar size, |χ| ∝ Z. It is negative, small (about 10⁻⁵) and independent of temperature. Why all electrons contribute: Larmor precession is set up in every electron, whether or not its shell is full, because it depends only on e, m and B. In closed shells the permanent orbital and spin moments cancel, leaving only the induced term, and the induced moments of all electrons add.

(b) Debye theory of the specific heat of solids

Assumptions: the solid is an elastic continuum with 3N normal modes (N atoms) and ω = vk. The modes are standing waves with a frequency cut-off ω_D, fixed by counting 3N modes. Density of modes (3 polarisations, mean speed v): g(ω)dω = (3Vω²/2π²v³)dω. Setting ∫₀^ω_D g dω = 3N gives ω_D³ = 6π²Nv³/V, so g(ω) = 9Nω²/ω_D³. Energy: each mode has mean energy ħω/(e^(ħω/kT) − 1), so U = ∫₀^ω_D [ħω/(e^(ħω/kT) − 1)] g(ω)dω. With x = ħω/kT and θ_D = ħω_D/k: C_V = dU/dT = 9Nk(T/θ_D)³ ∫₀^(θ_D/T) x⁴eˣ/(eˣ − 1)² dx. High T (T ≫ θ_D): eˣ − 1 ≈ x, the integral ≈ (θ_D/T)³/3, so C_V = 3Nk (Dulong-Petit). Low T (T ≪ θ_D): the upper limit tends to ∞ and the integral to 4π⁴/15, so C_V = (12π⁴/5)Nk(T/θ_D)³ ∝ T³. Agreement: measurements follow the universal curve of C_V/3Nk against T/θ_D; the T³ law holds for T < θ_D/50 (Cu θ_D ≈ 343 K, Pb 105 K, diamond 2230 K) and C_V approaches 3R at high T. Key assumption versus Einstein: a continuous spectrum of frequencies up to ω_D, instead of one common frequency. This gives the T³ fall-off, whereas Einstein's model gives an exponential fall, below the observed values. Drawback: it treats the solid as a continuum with one mean sound speed and a sharp cut-off, ignoring optical branches and the real phonon density of states, so θ_D varies slightly with temperature; for metals, the electronic term γT must be added.

(c) Wien-bridge oscillator

Circuit: an amplifier (op-amp or two-stage transistor amplifier) has two feedback paths. The positive-feedback path is a Wien network: R₁ in series with C₁, and R₂ in parallel with C₂, from the output to the non-inverting input. The negative-feedback path is a resistive divider R₃ (output to the inverting input) and R₄ (inverting input to ground). Working: with R₁ = R₂ = R and C₁ = C₂ = C, the fraction fed back is β = Z₂/(Z₁ + Z₂), where Z₁ = R + 1/jωC and Z₂ = R/(1 + jωRC). It is real, so the phase shift is zero, only when ωRC = 1, and then β = 1/3. Hence the oscillation frequency is f = 1/(2πRC). Below that frequency the network gives a phase lead, and above it a phase lag, so the Barkhausen phase condition holds only at f. Amplitude condition: the loop gain Aβ must equal 1, so the amplifier gain A = 1 + R₃/R₄ must be 3, i.e. R₃ = 2R₄. Starting from noise, a gain slightly above 3 lets the oscillation grow. To stabilise the amplitude, R₄ is a lamp with a positive temperature coefficient (or diodes limit the signal), so that as the amplitude rises, the gain falls back to 3. The output is a stable sine wave, useful for audio frequencies (for example 20 Hz to 20 kHz) with R and C ganged variable.