Noether Theorem - Classical Mechanics - Past Paper, Exams of Classical Mechanics

Download Noether Theorem - Classical Mechanics - Past Paper and more Exams Classical Mechanics in PDF only on Docsity! L A N C A S T E R U N I V E R S I T Y 2012 EXAMINATIONS Part II PHYSICS - Paper 2.F • Candidates should answer all those sections identified with the modules for which they are registered. • The time allocated is 60 minutes per section. • An indication of mark weighting is given by the numbers in square brackets following each part. • In each section attempted, candidates should answer question 1 (30 marks) and either question 2 or question 3 (30 Marks). • Use a separate answer book for each section. PHYSICAL CONSTANTS Planck’s constant h = 6.63× 10−34 J s ~ = 1.05× 10−34 J s Boltzmann’s constant kB = 1.38× 10−23 JK−1 Mass of electron me = 9.11× 10−31 kg Mass of proton mp = 1.67× 10−27 kg Electronic charge e = 1.60× 10−19C Speed of light c = 3.00× 108ms−1 Avogadro’s number NA = 6.02× 1023mol−1 Permittivity of the vacuum ²0 = 8.85× 10−12 Fm−1 Permeability of the vacuum µ0 = 4π × 10−7Hm−1 Gravitational constant G = 6.67× 10−11Nm2 kg−2 Bohr magneton µB = 9.27× 10−24 JT−1 (or Am2) Bohr radius a0 = 5.29× 10−11m Gas constant R = 8.31 JK−1mol−1 Acceleration due to gravity g = 9.81m s−2 1 standard atmosphere = 1.01× 105Nm−2 Mass of Earth = 5.97× 1024 kg Radius of Earth = 6.38× 106m please turn over 1 Section A: Module 273: Classical Mechanics (The time allocated for this section is 60 minutes. Candidates should answer question A1 and either question A2 or question A3.) Compulsory question: A1. (a) State the least action principle for a particle described by a Lagrangian L(x, ẋ, t). [5] (b) A non-relativistic particle of mass m is moving in one dimension under the influence of a constant force F . (i) State the Lagrangian for this particle. (ii) Consider a trajectory x(t) = x0 + v0t+ αt 2. Use the boundary conditions x(0) = x(T ) = 0 to express x0 and v0 in terms of α and T . (iii) Calculate the action for this trajectory as a function of α. (iv) Find the value of α for which the action is minimal. [10] (c) (i) State Noether’s theorem for a Lagrangian possessing a continuous symme- try. (ii) Show that the Lagrangian L(x, y, ẋ, ẏ) = ẋ2 + ẏ 2 y2 + exy is invariant under the transformation x → x+ λ, y → e−λy where λ is a continuous parameter. (iii) Use Noether’s theorem to derive the corresponding conserved quantity. (iv) Determine the mechanical energy for this Lagrangian, and explain why this is a conserved quantity. [15] 2 Section B: Module 274: Classical Fields (The time allocated for this section is 60 minutes. Candidates should answer question B1 and either question B2 or question B3.) Compulsory question: B1. (a) Derive the homogeneous Maxwell equations ∇ ·B = 0 and ∇× E+ ∂ ∂t B = 0 in integral form, using Gauss’ theorem and Stokes’ theorem. [6] (b) Explain what is meant by the TE, TM and TEM propagating modes in a waveguide. [3] (c) (i) Find an expression for the scalar potential V (r) for the electrostatic field E(r) = Axx̂+Byŷ+ Czẑ where A, B, and C are constants. (ii) State the shape of the surfaces of constant V for the case that A, B, and C are positive and equal A = B = C. [8] (d) Consider a neutral plasma consisting of sheets of electrons with number density ne and protons with number density ni. The sheets are initially separated by a distance ∆x and both the electrons and the protons are free to move independently along the x-axis only. + + + _ _ + + + + _ _ _ x∆x _ _ Electrons _ Protons (i) State the equations of motion for the electrons and for the protons. (ii) Show that the natural oscillation frequency ωp of the system is determined by the expression ω2p = ω 2 pe + ω 2 pi where ωpe is the electron oscillation frequency and ωpi is the proton oscil- lation frequency. [13] please turn over 5 Answer one of the following two questions: B2. (a) (i) Briefly discuss the term retarded potentials. (ii) Sketch a diagram showing the position of a point charge q at time t moving along an arbitrary trajectory in free space. Indicate in your diagram the vector to a field point r, the vector showing the retarded position w, and the vector R from the retarded position to the field point. (iii) Write down an expression for the retarded time tr in terms of r and t. (iv) Briefly discuss the consequences of retarded potentials in terms of the observed appearance of moving objects. [12] (b) The electric field of a point charge q moving with velocity v and acceleration a is given by the expression E(r, t) = q 4π²0 R (R · u)3 [ (c2 − v2)u+R× (u× a)] where u = cR̂− v (i) Identify in the above expression the velocity field and the acceleration field and state their physical meanings. (ii) Derive an expression for the associated magnetic field. [10] (c) (i) Use the information in part (b) to derive an equation for the force on a test charge Q moving with velocity V. (ii) Discuss the significance of this equation. [8] 6 B3. (a) Use the differential form of Gauss’ Law and the Ampère-Maxwell Law to derive the expression for charge conservation ∂ ∂t ρ+∇ · J = 0 [6] (b) (i) Find the electric field E and the magnetic field B if the vector potential A and the scalar potential V are given by the expressions A(r, t) = A0[cos(kz − ωt)x̂+ sin(kz − ωt)ŷ] and V (r, t) = 0 (ii) Calculate the Poynting vector S assuming the medium has a constant relative permittivity µr. [10] (c) (i) Briefly discuss the concept of a gauge transformation in terms of the un- derlying physical principles. (ii) Write down expressions for a general gauge transformation of the vector potential A and the scalar potential V . (iii) State the Coulomb Gauge and the Lorentz Gauge in terms of V and A. (iv) Use the expressions ∇2V + ∂ ∂t (∇ ·A) = − 1 ²0 ρ and −µ0J = ( ∇2A− µ0²0∂ 2A ∂t2 ) −∇ ( ∇ ·A+ µ0²0∂V ∂t ) along with the Lorentz Gauge to derive uncoupled equations for the vector potential A and the scalar potential V . [14] please turn over 7