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

Part (a): Classical diamagnetism. In a magnetic field B, each bound electron’s orbital motion is slightly altered by the Lorentz force. The orbit acquires a Larmor precession about the field axis with angular velocity Ω = −eB/2mₑ. For an electron, the induced magnetic moment is opposite to B. Using the perpendicular radius ρ to the field and averaging over an isotropic atom, ⟨ρ²⟩ = (2/3)⟨r²⟩, the change in moment per electron is Δμ = −(e²B/6mₑ)⟨r²⟩. For a neutral atom with Z electrons, Δμ_atom = −(e²B/6mₑ)Σᵢ⟨rᵢ²⟩. If N atoms per unit volume, M = NΔμ_atom and B = μ0H, so χ = M/H = −μ0N e²/(6mₑ)Σᵢ⟨rᵢ²⟩. For comparable mean-square radii, Σᵢ⟨rᵢ²⟩ ≈ Z⟨r²⟩, giving χ = −μ0NZ e²⟨r²⟩/(6mₑ). The negative sign is the defining feature of diamagnetism, and for similar atoms the magnitude is directly proportional to Z. All electrons contribute because the Larmor term depends only on charge, mass and orbital radius, not on the shell’s permanent moment; in closed shells permanent orbital and spin moments cancel, so the induced moments of every electron add rather than cancel. Even in partially filled shells, the induced term exists for every electron, though paramagnetism may mask it.

Part (b): Debye theory. Debye treats the solid as 3N harmonic oscillators whose low-frequency acoustic phonons have linear dispersion ω = vₛ k. The density of states is taken as g(ω)=9Nω²/ω_D³, normalized so ∫0^ωD g(ω)dω=3N. The thermal energy is U=∫0^ωD ℏω/(e^ℏω/kT−1)g(ω)dω. Differentiating, C_V=dU/dT. With x=ℏω/kT and θ_D=ℏω_D/k, C_V=9Nk_B(T/θ_D)³∫0^θ_D/T x⁴e^x/(e^x−1)²dx; for one mole, replace 9Nk_B by 9R. At high T, θ_D/T≪1; expanding the integrand gives C_V≈3Nk_B=3R, the Dulong–Petit law. At low T, θ_D/T≫1; the integral tends to 4π⁴/15, so C_V≈(12π⁴/5)Nk_B(T/θ_D)³, the T³ law. The integral is the Debye function, and a universal plot of C_V/3R against T/θ_D matches data. Copper, lead and diamond show a linear C_V versus T³ region at low temperatures, from which θ_D is extracted (about 340 K for Cu, 105 K for Pb, 2200 K for diamond), and approach 3R at high T. The key assumption, unlike Einstein’s single-frequency model, is a continuous acoustic-phonon spectrum up to a cutoff, which supplies the missing low-frequency modes; Einstein’s model instead gives an exponential low-temperature fall-off. Its drawbacks are the isotropic-solid and linear-dispersion assumptions, use of one average sound velocity, and neglect of optical phonons, so it is not exact at intermediate temperatures or in crystals with strong optical modes.

Part (c): Wien-Bridge oscillator. The circuit is a bridge whose two arms are frequency-dependent: Z1 is R1 in series with C1, and Z2 is R2 in parallel with C2; the other two arms are resistors R3 and R4. In the diagram, the amplifier output drives the bridge nodes, and the junction of R3 and R4 returns to the non-inverting input. At balance, Z1Z4=Z2Z3. For R1=R2=R and C1=C2=C, this condition is satisfied at ω=1/RC, so f=1/(2πRC), and the bridge output is in phase with the input. The phase shifts of the two RC arms make the bridge output in phase with the input only at this frequency, so the Barkhausen phase condition is met only at the Wien frequency. The voltage division in the resistive arms gives a feedback fraction of 1/3 when R3/R4=2, so the amplifier must have a gain of 3; this is the Barkhausen amplitude condition. In practice the gain is made slightly larger than 3 to start oscillations. As the output grows, a nonlinear element such as an incandescent lamp or diodes in the feedback path changes its resistance, reducing the loop gain to unity and stabilizing the amplitude. The result is a sustained sinusoid at f=1/(2πRC). Thus the explanations rest on clear causal chains: field-induced Larmor precession, the Debye acoustic-phonon spectrum, and frequency-selective positive feedback.