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Derive the continuity equation for fluid transport |
Derive the continuity equation for fluid transport |
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− | :<math> frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0 </math>. |
+ | :<math> \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho v) = 0 </math>. |

## Revision as of 07:39, 14 November 2018

## Contents

**Section 1: Mechanics**

**1. The skydiver**

Consider a skydiver falling out of an airplane. Approximate the external force on the skydiver, given that air resistance is proportional to the velocity squared .

a) Use the force equation to calculate the terminal velocity .

b) Change your equation from the previous part into a first-order differential equation. Then use your result to arrive at the integral

- .

c) Solve the above integral. Then rearrange the velocity solution as a function of time.

**2. The spinning bucket**

Consider a bucket of water rotated at the z-axis as pictured below. Suppose the water rotates around the z-axis at some rotation rate . Show that the depression of the water surface is a parabola. Find the potential energy of the effective force.

**3. Mechanics of the Coulomb potential**

Consider the problem of firing a proton at a nucleus of atomic number (so that one views the nucleus as immobile and the origin of the coordinate system, only the proton moves). Work in spherical polar coordinates.

a) Write the Lagrangian for the interaction, using your knowledge of the electrostatic forces.

b) Determine the Euler-Lagrange equations of motion of the proton. What is the conserved quantity?

**4. The continuity equation**

Derive the continuity equation for fluid transport

- .

**Section 2: Electromagnetism**

**1. Gauss’ law for magnetism**

First show that ∇∙(∇×A)=0 for a vector field. The magnetic field is the curl of the vector potential B=∇×A. Use this vector property to derive Gauss’ law for magnetism ∇∙B=0.

**2. Maxwell’s extension**
Figure out why ∇×B=μ_0 J is not the complete picture, and derive the Ampere-Maxwell law for time-varying EM fields
∇×B=μ_0 J+〖μ_0 ϵ〗_0 ∂E/∂t.

**3. EM cycloid motion**

Consider a charged particle subjected to the initial conditions E=(0,0,E) B=(B,0,0) v_0=(0,y ̇,z ̇ )=(0,0,0). Determine the trajectory r=(0,y(t),z(t)) if the charged particle is released from the origin. This is actually a 2-D problem, but the cross products work in 3-D; hence, the x-components are set to zero.

**4. Energy transport of EM fields**

Starting with J=1/μ_0 ∇×B-ϵ_0 ∂E/∂t and the energy density transport d/dt ∫▒u dV=∫▒〖E∙J〗 dV derive Poynting’s theorem in the form -∂u/∂t=∂/∂t (1/2 ϵ_0 E^2+1/(2μ_0 ) B^2 )+∇∙S.

**Section 3: Thermal physics**

**1. Adiabatic expansion**

Derive the formula for an adiabatic expansion of ideal gas PV^γ=const, where γ=C_P/C_V is a constant. Start from the first law of thermodynamics dU=dQ-dW, the ideal gas law PV=nRT, and the relation C_P-C_V=R.

**2. Boxes of ideal gas**

Consider two boxes of volume V each holding N atoms of ideal gas. Both boxes are initially isolated from each other and the surroundings. Box 1 and box 2 have initial temperatures T_1 and T_2 respectively. The boxes are then brought into thermal contact. a) Calculate the equilibrium temperature of the two boxes. b) Calculate the entropy change using S=Nk_B ln(E^(3/2) V) and explain why ΔS≥0.

**3. Statistical physics of protein folding**

For a simple lattice model of protein folding, the residues are connected in links. Consider a four-residue protein model. The open configurations cost no energy, and the folded configurations reduces the energy by E=ϵ. a) If there are 36 configurations in total, where 8 configurations are folded, calculate the partition function. Then write expressions for the probability when the protein is open and when the protein is folded. b) What is the probability of the protein being in the folded state when the temperature is large (T→∞)? c) Calculate the average energy of the protein.

**4. Derivation of the heat equation**

In thermohydraulics, energy conservation is encapsulated in the vector field differential form ∂ϵ/∂t+∇∙(ϵv)+∇∙q_c=a. Let the energy density be ϵ=ρcT, where the specific heat c is constant. The conductive heat transfer follows Fourier’s law q_c=-k∇T, where k is constant. Derive the heat equation ρc ∂T/∂t+ρcv∙∇T-k∇^2 T=a. Hint: you will need to use the conservation of mass ∂ρ/∂t+∇∙(ρv)=0.

**Section 4: Special relativity**

**1. Relativistic momentum-energy relation**

Begin with the relativistic momentum and energy: p=mv/√(1-v^2/c^2 ) E=(mc^2)/√(1-v^2/c^2 ) Derive the relativistic energy-momentum relation: E^2=(pc)^2+(mc^2 )^2

**2. Lorentz transformation matrix**

Given the Lorentz transformation in one direction x^'=γ(x-vt) t^'=γ(t-vx/c^2 ) expressed the above set of equations as a 2x2 matrix. Calculate the determinant and find the inverse matrix.

**3. Muon decay**

What is the minimum energy needed for a muon, generated by interactions of cosmic rays with the upper atmosphere, to reach sea level? The thickness of the atmosphere is roughly 10 km. The lifetime in the muon’s rest frame is 2.2μs. The rest mass of a muon is 105.658 MeV/c^2.

**4. Relativistic Doppler effect**

Begin with the wavelength λ=cΔt-vΔt and frequency of the source f_s=1/Δt' to derive the relativistic Doppler effect for the frequency measured by the observer of an approaching source f_obs=√((1+β)/(1-β)) f_s. Note that β=v/c.

**Section 5: Quantum mechanics**

**1. Ehrenfest theorem and the Commutator**

Consider this important identity d〈A ̂ 〉/dt=i/ℏ 〈[H ̂,A ̂ ]〉+〈(∂A ̂)/∂t〉. Prove the Ehrenfest theorem d〈p ̂ 〉/dt=-〈(∂V ̂)/∂x〉.

**2. Step potential**

A beam of particles with kinetic energy E coming from the left encounters a step barrier described by V(x)={█(0,&x<0@V,&x≥0)┤ Find the probability of transmission and reflection if E>V.

**3. Hydrogen-like atom**

Verify that the n=1,l=0,m=0 one-electron wavefunction ϕ_100 (r)=Ae^(-Zr/a_B ) satisfies the Schrodinger equation in spherical polar coordinates with a Coulomb potential [-ℏ^2/2m 1/r^2 d/dr (r^2 d/dr)-(Ze^2)/(4πϵ_0 r) ]ψ=Eψ. Note that the Bohr radius is defined a_B=(4πϵ_0 ℏ^2)/(m_e e^2 ) where m_e is the rest mass of an electron. Then calculate the normalization constant A.

**4. Eigenstates of a Hamiltonian**

Consider a system with two linearly independent states |├ 1⟩┤=(1¦0) and |├ 2⟩┤=(0¦1) . The system is described by the Hamiltonian H=ϵ(■(0&i@-i&0)) where ϵ is a positive real number. a) Find the eigenenergies and the normalized eigenfunctions of H. b) If |├ Ψ(0)⟩┤=|├ 1⟩┤, find |├ Ψ(t)⟩┤ for t>0. c) Consider an observable Q ̂=μ(├|├ 1⟩┤ ⟨├ 1┤|┤-├|├ 2⟩┤ ⟨├ 2┤|┤). Find 〈Q ̂ 〉 as a function of time in the state |├ Ψ(t)⟩┤.