Theoretische Physik 2: Elektrodynamik (Prof. A-S. Smith) Tutorial 12

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1 WiSe Prof. Dr. A-S. Smith Dip.-Phys. Een Fischermeier Dip.-Phys. Matthias Saba am Lehrstuh für Theoretische Physik I Department für Physik Friedrich-Aexander-Universität Erangen-Nürnberg Probem 12.1 Theoretische Physik 2: Eektrodynamik Prof. A-S. Smith) Tutoria 12 Point charge and dieectric sphere Consider a point charge Q in front of a dieectric sphere of permittivity ε and radius R. The distance between the origin of the sphere and the charge is d. a) The eectric fied may be written in terms of a scaar potentia E r) = Φ r) that satisfies the Lapace equation everywhere except for the position of the point charge and the surface of the sphere. Why? b) For convenience, we choose the coordinate frame so that the charge is ocated on the z-axis see figure). Due to axia symmetry of the probem, the potentia exhibits the genera form: Φ r) = a r + b r +1)) P cos θ) Derive the ansatz: =0 Φ r) = a r, r R P cos θ) Qd +1) r + b r +1)), R < r < d Qd ) + c r +1), r > d Q d e z R ε Hint: The potentia is buid up by the point charge and induced charges inside the sphere. Expand each contribution with 1 + x 2 2xt) 1/2 = P t)x for x < 1 and t 1 c) State the matching conditions at r = R and r = d. To find the appropriate matching condition near the charge singuarity at r = d, consider a singuar charge distribution ρx) = δx)σ 0. Cacuate the eectric fied and show that it is not continuous at the origin. Convince yoursef that it is possibe and suitabe) to gauge the potentia to be continuous at the singuarity. That is true for any interface. d) Cacuate the coefficients: a = Q 1 + ε + 1) d d ) 1 ε R b = c = R +1 Q 1 + ε + 1) d d

2 Probem 12.2 Sound waves in fuid The macroscopic properties of a fuid are characterized in terms of a few fieds, e.g., the mass density ρ r, t), the mass current density j r, t), the fuid veocity v r, t), and the pressure p r, t). Euer s equations specify the fied equations; the first set encodes the conservation of mass and momentum, t ρ + k j k = 0, t j k + Π k = 0. ) The mass current density is connected to the fuid veocity by j r, t) = ρ r, t) v r, t), and Π k denotes the momentum current tensor Π k = ρv k v σ k = ρv k v + pδ k, ) which coses the equations. The term ρv k v is the contribution to the momentum current by the inertia of the fow the terms responsibe for turbuence). The quantity σ k = pδ k + σ k is known as the stress tensor. p denotes the pressure whie σ k encompasses the buk and shear) viscous forces, i.e. dissipative processes, which are negected in Euer s equations, σ k = 0. a) Demonstrate that ρ r, t) = ρ 0 = const, v r, t) = 0, and p r, t) = p 0 = const constitutes a soution of the fied equations. Show that the inearized fied equations for sma perturbations δρ = ρ ρ 0, v, and δp = p p 0 to this reference state read t δρ + ρ 0 k v k = 0, ρ 0 t v k = k δp. Introduce the isotherma compressibiity κ T that refects the pressure increase due to compression at constant temperature to inear order, δp = δρ/. b) Derive a oca conservation aw, t u + S = 0, for the energy density u r, t) = ρ 0 2 v r, t)2 + A δρ r, t)2 2 for suitaby chosen A reying on the approximations introduced so far. Determine the energy current density S x, t). c) Show that the inearized fied equations aow for monochromatic ongitudina waves in v and scaar waves in δρ r, t). Probem 12.3 Drude-Ha mode As an extension of Drude s theory of conductors consider the induced current density j ind) r, t) in the presence of a constant and uniform externa magnetic fied B = Bê z. Motivate the constitutive equation t j ind) r, t) + 1 τ j ind) r, t) e m c B j ind) r, t) = ω2 p 4π E r, t), ω 2 p = 4πne2 m, where m denotes the effective mass of the conduction eectrons charge e), τ a characteristic reaxation time, and n the density of conduction eectrons. The characteristic frequency ω p is referred to as pasma frequency. a) Perform a tempora Fourier transform, convention E r, ω) = e iωt E r, t)dt. Show that the response becomes oca in the frequency domain, j ind) k r, ω) = σ k ω) E r, ω), and determine the dynamic magneto-conductivity tensor σ k ω). b) Speciaize to d.c. fieds, i.e. ω = 0, and discuss the Ha resistivity. Due date:

3 WiSe Prof. Dr. A-S. Smith Dip.-Phys. Een Fischermeier Dip.-Phys. Matthias Saba am Lehrstuh für Theoretische Physik I Department für Physik Friedrich-Aexander-Universität Erangen-Nürnberg Theoretische Physik 2: Eektrodynamik Prof. A-S. Smith) Soutions to Tutoria 12 Soution of Probem 12.1 Point charge and dieectric sphere a) Everywhere outside the sphere and not at the point charge Gauss s aw reads E = 0 impying Φ = 0 by definition. Everywhere inside the sphere it reads ε D) = 0 and as ε is constant the divergence aw hods for E as we and Φ = 0 foows. b) Inside the sphere incudes the point r = 0 and contributions in r +1 woud diverge. A genera soutions is given by: Φ = P cos θ)a r Outside of the sphere but coser to the origin, the potentia may be expanded in terms of the point charge Q and the induced axiay symmetric charge distribution inside the sphere ρ ind actuay, it can ony be a surface charge density but for convenience we write it as a voume density): Q Φ = r d e z + d 3 x ρ ind x) x R x r The point charge part may be expanded by: Q d 1 + r/d) 2 2r/d) cos θ = Q d r P cos θ) d) And the charge density by: d 3 ρ x ind x) r 1 + x/r) 2 x/r) cos γ = d 3 x ρ ind x) r x ) P cos γ) r Hence ony contributions in r +1) come from the induced charges. The foowing forma derivation is given here for the sake of competeness but is not expected from the students: Note that γ is the ange between x and r. The charge density may aso be expanded by ρ ind x) =

4 ρ x)p cos θ x) = ρ x)y,0 x)n, = = =,,m r +1) dx x ρ x) r +1) Y m r) 4πN r +1) 4πN Y 0 r) }{{} P cos θ)/n 4π r +1) P cos θ) and we obtain: dω x N Y,0 x) P cos γ) }{{} dx x +2 ρ x) dx x +2 ρ x) dx ρ x)x +2 } {{ } :=b = m 4π/2+1)Y m r)y m x) dω x Y 0 x)y m x) } {{ } δ, δm0 We may hence write for the potentia at R < r < d: Φ = P cos θ) Qd +1) r + b r +1)) In the outside region r > d we expand the monopoe in d/r and obtain with the condition that Φ 0 at infinity): Φ = ) P cos θ) Qd + c r +1) c) The matching conditions at R = r are that the transverse part of E and the norma part of D are continuous. That incudes for the potentia for δ 0 is Φr + δ) = Φr δ) and εφ R δ) = Φ R + δ) In case of the one dimensiona probem the E-fied is obtained by integration over x: { σ 0 /2, x < 0 E = σ 0 /2, x > 0 The potentia is obtained by another integration to Φ = x σ/2 if we choose the gauge im x 0 = 0 in both regions. The potentia cannot vary in y or z due to symmetry. Likewise one argues for an arbitrary differentiabe boundary that the potentia can be chosen continuous as the transversa fied components are continuous and hence the moduation of Φ aong the surface. For this probem that is Φ is continuous at r = d. d) The matching conditions at r = R read: The soution of this set of equations is: a R = QR d +1) + b R +1) εa R 1 = Qd +1) R 1 + 1)b R +2) a = Q 1 + ε + 1) d d ) 1 ε R b = R +1 Q 1 + ε + 1) d d The part of the potentia coming from the charge Q monopoe without sphere) is continuous at r = d so that the remaining part aso has to be continuous and c = b. 2

5 Soution of Probem 12.2 Sound waves in fuid Wave phenomena were known ong before the works by Maxwe and Hertz, one exampe are sound waves. An essentia difference to eectrodynamic waves is the existence of ongitudina waves. The mass conservation aw is competey anaogous to the charge conservation aw of eectrodynamics, t ρ + j = 0. The mass current density is connected to the fuid veocity by j r, t) = ρ r, t) v r, t). In this form it is cear momentum = mass veocity) that j r, t) constitutes aso the momentum density. A conservation aw for the momentum hods, t j k = Π k, where Π k is the momentum current tensor simiar to Maxwe s momentum current tensor). Paying a itte bit with Gaieian invariance shows that Π k = ρv k v σ k = ρv k v + pδ k σ k The term ρv k v is the contribution to momentum current by free fow these terms are responsibe for turbuence). The quantity σ k = pδ k + σ k is known as stress tensor, p denotes the pressure. Last σ k encodes the buk and shear) viscous forces. In Euer s equations, dissipative processes are negected, σ k = 0. a) The fied equations ) and ) are obviousy satisfied by ρ r, t) = ρ 0 = const, v r, t) = 0, and p r, t) = p 0 = const. In inear order, it hods j = ρ 0 v δρ v is of second order in the perturbations), mass and momentum conservation yied t δρ + ρ 0 k v k = 0 and ρ 0 t v k = k δp. At constant temperature, the pressure increases upon compression. In the regime of inear response, this behavior is described by the isotherma compressibiity κ T, δp = 1 δρ, or as thermodynamic derivative, κ T = 1 ) ρ. ρ 0 p Then the equations of motion read in inear order t δρ = ρ 0 k v k, ρ 0 t v k = 1 k δρ. b) The energy density is quadratic to owest non-trivia order and consists of the kinetic energy of the fuid and the energy of compression simiar to a Hookean spring), u r, t) = ρ 0 2 v r, t)2 + A 2 δρ r, t)2. A conservation aw is derived by cacuating the derivative with respect to time, t u = ρ 0 v k t v k + Aδρ t δρ = v k k δρ Aδρ ρ 0 k v k = k [ ] 1 v k δρ, T 3

6 provided A = 1/ρ 2 0 κ T. Defining the isotherma sound veocity c := 1 ρ0 κ T, the energy density and the energy current density are defined as u r, t) = ρ 0 2 v r, t)2 + c2 2ρ 0 δρ r, t) 2 and S = c 2 v δρ. Then the energy conservation aw for the fuid reads t u + div S = 0. Note the cose anaogy with the Poynting vector and the energy conservation aw in eectrodynamics. c) The wave equations are derived by taking tempora and spatia derivatives of the inearized fied equations. For density waves, t 2 δρ = ρ 0 t v } ) 1 ρ 0 t 1 v = 2 δρ c 2 2 t 2 δρ = 0. This aows for monochromatic scaar waves, Simiary, δρ r, t) = ˆρ k) e i k r ωt) + c.c. with the dispersion reation ω = ck. t δρ = ρ0 v) } ρ 0 t 2 v = 1 t δρ 1 c 2 2 t v = v). Spit the veocity fied in a soenoida and a cur-free part, v = v + v such that v = 0 and v = 0. Then, there exists some scaar fied ϕ x, t) with v = ϕ and we have v) = v ) ϕ ) = 2 v. We concude a wave equation for v, ) 1 c 2 2 t 2 v r, t) = 0, with ongitudina waves as soution, v r, t) = ˆv k) ˆ k e i k r ωt) + c.c. ) with the unit vector ˆ k = k/ k. The transverse part obeys 2 t v = 0 ; in particuar, there are no transverse waves in our ideaized fuid. Aternativey, you may use the Ansatz of monochromatic waves this corresponds to a Fourier transform of the probem), v r, t) = v k) e i k r ωt), and define v = k v) k/k 2 and v = k v) k/k 2. Then, the wave equation is satisfied for the ongitudina parts v, provided ω = ck, and has no soution for the transversa part v. Remark: For thermay isoated systems the true ong-waveength hydrodynamic modes are sound waves with the adiabatic sound veocity. 4

7 Soution of Probem 12.3 Drude-Ha mode We start with Newton s equations for a singe charge in externa eectric E r, t) and magnetic fied B, m vt) + γ vt) = e) E e) + c v B v + 1 τ v = e m E e m c v B. With the current density j ind) = n e) v, we introduce the constitutive equation t j ind) r, t) + 1 τ j ind) r, t) e m c B j ind) r, t) = ω2 p 4π E r, t) Stricty speaking, the eectric fied is spatiay varying as the eectrons move. Impicity, there is the assumption that E r, t) is sowy varying in space. The response is then oca in space. The dependence on r wi be dropped in the foowing as we as the superscript ind). end of preiminary remarks a) A tempora Fourier transform, Eω) = e iωt Et)dt, makes the response in the frequency domain oca. The time derivative transates in the Fourier space to a mutipication, e iωt t jt)dt = e iωt jt) iω e iωt jt)dt = iω jω). t= Then, or in components, iω + 1 ) jω) e τ m B c jω) = ω2 p Eω), 4π iω + 1 ) j x ω) + ω c j y ω) = ω2 p τ iω + 1 ) j y ω) ω c j x ω) = ω2 p τ 4π E xω) 4π E yω) iω + 1 ) j z ω) = ω2 p τ 4π E zω) with the cycotron frequency ω c = eb/m c. The ast reation is inverted readiy, j z ω) = σ zz ω)e z ω), σ zz ω) = σ 0 1 iωτ with the static Drude conductivity σ 0 = ne 2 τ/m. As a consequence σ zx ω) = σ zy ω) = 0. The remaining two equations, ) ) ) 1 iωτ ωc τ jx ω) Ex ω) = σ ω c τ 1 iωτ j y ω) 0, E y ω) are easiy soved by matrix inversion, ) ) ) ) ) jx ω) σ 0 1 iωτ ωc τ Ex ω) σxx ω) σ = j y ω) 1 iωτ) 2 + ω c τ) 2 = xy ω) Ex ω). ω c τ 1 iωτ E y ω) σ yx ω) σ yy ω) E y ω) 5

8 Note the symmetry σ ik ω, B) = σ ki ω, B), where the dependence on the magnetic fied has been indicated expicity, in agreement with Onsager s reciproca reations. b) For d.c. currents, i.e., at ow frequencies ωτ 1, the conductivity tensor simpifies to σω = 0) = σ ω c τ) 2 ) 1 ωc τ. ω c τ 1 The diagona eements describe the d.c. conductivities which decrease with increasing magnetic fied. This effect of magneto-resistance is interpreted as foows: the eectrons compete many Larmor cyces and foow the E B drift before a scattering event occurs. However on these orbits, there is no net contribution to the current aong the fied. At the same time, the the E B drift becomes arger and the Ha effect buids up. A current in x-direction j y = 0) yieds a potentia drop U H aong the y-axis, U H = E y d = ω cτ σ 0 d j x = eb/m c τ ne 2 τ/m d j x = B/c ne d j x. where the Ha votage is measured on a pate of width d. The Ha effect provides a convenient too for measuring the strength of a magnetic fied since U H is independent of the resistance of the meta and depends merey on the Ha coefficient, R H := U Hd j x B/c = 1 ne. At ow temperatures and for strong magnetic fieds, quantum effects become reevant. Then, ony a sma number of conduction eectrons contribute to the Ha votage resuting in a discretization of R H. Nowadays, a singe such quantum, h/e Ω, serves as unit for the eectrica resistance. 6