Category: Part 2: Thermodynamics

Problem Statement Solve the kinematics problem: Solve the kinematics problem: Show that the mean square displacement of a Brownian particle grows as $\langle x^2\rangle = 2Dt$. Estimate $D$ for a sphere of radius $r=1\ \mu\text{m}$ in water at $T=300\ \text{K}$. From the diffusion equation $\partial n/\partial t = D\partial^2 n/\partial x^2$, a p Given Information…

Problem Statement State the fluctuation-dissipation theorem and explain its physical content. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and…

Problem Statement Solve the nuclear physics problem: Derive the Einstein relation $D = \mu_e k_BT$ linking the diffusion coefficient $D$ and the mobility $\mu_e$ of Brownian particles. All quantities, constants, and constraints stated in the problem above Physical constants used as needed (see Concepts section) This problem draws on fundamental physic Given Information See problem…

Problem Statement Solve the thermodynamics problem: Solve the thermodynamics problem: A rubber band warms when stretched adiabatically. Explain this using thermodynamics and the statistical view of entropy. Statistical: Unstretched rubber has polymer chains in high-entropy random coil configurations ($\Omega$ large). Stretching aligns the chains, red Given Information See problem statement for all given quantities. Physical…

Problem Statement A resistor $R=1\ \text{k}\Omega$ is at $T=300\ \text{K}$. Find the rms thermal voltage noise in a bandwidth $\Delta f=10\ \text{kHz}$. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations…

Problem Statement Solve the magnetic field/force problem: Solve the magnetic field/force problem: Write the first law for a paramagnetic solid including magnetic work, and derive the analogue of the ideal gas law. For a magnetic system in an external field $H$, the magnetic moment $M$ (total) does work $dW_{mag} = -\mu_0 H\,dM$ (work done by…

Problem Statement Solve the thermodynamics problem: Solve the thermodynamics problem: Show that adiabatic demagnetisation cools a paramagnet. If $M = CH/T$ (Curie law) and the lattice heat capacity is $C_{lat}$, find the temperature drop when $H$ is reduced from $H_1$ to $H_2$. The entropy of an ideal paramagnet: $S_{mag} = f(H/T)$ (function of $H/T$ Given…

Problem Statement Solve the fluid mechanics problem: Solve the fluid mechanics problem: A liquid has volumetric expansion coefficient $\alpha_V = 2.1\times10^{-4}\ \text{K}^{-1}$ and isothermal compressibility $\kappa_T = 4.5\times10^{-10}\ \text{Pa}^{-1}$. Find the pressure needed to maintain constant volume when temperature rises by $\Delta T = 10\ Given Information See problem statement for all given quantities. Physical…

Problem Statement Solve the thermodynamics problem: Solve the thermodynamics problem: State the Debye $T^3$ law for the heat capacity of a solid at low temperature. The Debye model treats lattice vibrations as a spectrum of acoustic phonons with maximum frequency $\omega_D$ (Debye frequency). At low temperatures ($T \ll \theta_D = \hbar\omega_D/k_B$) Given Information See problem…

Problem Statement Define the spreading coefficient $S = \sigma_{SG}-\sigma_{SL}-\sigma_{LG}$ and state the conditions for spreading, partial wetting, and non-wetting. Given Information See problem statement for all given quantities. Physical Concepts & Formulas This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then…