Unformatted text preview: HW3 Phys5405 f03 due 9/18/03
1) In the notes we saw that the electrostatic ﬁeld changes discontinuously as you pass
through a layer of charge, while the potential is continuous. JDJ uses up alot of space
showing that the potential changes discontinuously when you pass through a dipole layer
of charges. We’ll investigate this in an easy way. Let the volume charge density be ρ(r) =
σ0 δ (r − R) + σ1 δ (r − R ), where σ0 , R, σ1 , R are constants and R < R . Find E, Φ for
a)r < R, R < r < R , r > R . Specialize to the case σ1 = −σ0 and R = R + ∆, ∆ << 1.
Now calculate Φ(r > R ) and Φ(r < R) and note the discontinuity.
2) An inﬁnitely long right cylindrical region of radius R has uniform volume charge density
ρ0 . It is surrounded by an uncharged coaxial right cylindrical conducting shell of inner radius
R’ (R’ > R) and outer radius R”. Find the E for a) ρ > R”, b) R < ρ < R’, c) ρ < R. Find
3) Show that the Schrodinger Equation, − 2m 2 Ψ = ih ∂ Ψ , is invariant under the Galillean
¯ ∂t transform, provided the wave function in the O’ frame is renormalized by a phase, Ψ = Ψeiα ,
where α = − mv (z − vt/2) and z is the direction of relative motion between O and O’. That
is, show that − 2m 2 Ψ = ih ∂ Ψ .
¯ ∂t 4) JDJ 11-4
5) Using the information about the relativistic Doppler shift show that the phase of an
electromagnetic wave k · r − ωt is a Lorentz Invariant (LI). Give a physical reason why this
is so. Thus kµ = (ω/c, ik) is a 4-vector. Here k = 2π/λ, ω/k = c in vacuum and the direction
of k is along the photon’s velocity. 1 ...
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