Electromagnetism - Lecture 15 Waves in Conductors Absorption in Conductors • Skin Depth • Reflection at Conducting Surfaces • Radiation Pressure • Power Dissipation • Conservation of Electromagnetic Energy • 1 Maxwell's Equations in Conductors Maxwell's Equations M1-3 are as for insulators: :D = ρ :B = 0 r C r @B E = r × − @t but M4 in conductors includes a free current density JC = σE: @D H = + JC r × @t @E B = µ + µ σE r × 0 r 0 @t 0 where we have again assumed that µr = 1 2 Solution of M1-4 in Conductors Taking the curl of M3: @ ( E) = ( B) = ( :E) 2E r × r × −@t r × r r − r using M1 with the assumption that the free charge density is zero: ρ = ( :E) = 0 r c r 0r r Subtituting for B from M4 leads to a modified wave equation: @ @2E @E 2E = ( B) = µ + µ σ r @t r × 0 r 0 @t2 0 @t The first order time derivative is proportional to the conductivity σ This acts as a damping term for waves in conductors 3 Notes: Diagrams: 4 Plane Waves in Conductors The solution can be written as an attenuated plane wave: i(!t−βz) −αz Ex = E0e e Substituting this back into the modified wave equation: ( iβ α)2E = µ (i!)2E + µ σ(i!)E − − 0 0 r 0 0 0 0 Equating the real and imaginary parts: β2 + α2 = µ !2 2βα = µ σ! − − 0 r 0 0 For a good conductor with σ !: r 0 µ σ! α = β = 0 r 2 For a perfect conductor σ there are no waves and E = 0 ! 1 5 Skin Depth The attenuation length in a conductor is known as the skin depth: 1 2 δ = = α rµ0σ! Skin depths for a good conductor (metal): δ 10cm at ν = 50Hz (mains frequency) • ≈ δ 10µm at ν = 50MHz (radio waves) • ≈ High frequency waves are rapidly attenuated in good conductors Practical application of this for RF shielding of sensitive equipment against external sources of EM waves. 6 Magnetic Field of Plane Wave The magnetic field of a plane wave in a good conductor can be found using M4: @H − − y = σE = σE ei(!t αz)e αz @z − x − 0 where we neglect the term in the wave equation proportional to r σE − − − − − H = 0 ei(!t αz)e αz = H ei(!t αz π=4)e αz y (1 + i)α 0 Note that there is a phase shift of π=4 between E and H The amplitude ratio depends on the ratio of σ and !: H σ 0 = E0 rµ0! In a good conductor H E and they are no longer related by c! 0 0 7 Notes: Diagrams: 8 Reflection at a Conducting Surface We just consider the case of normal incidence The incident wave has: − E − E = E ei(!t k1z)x^ B = 0I ei(!t k1z)y^ I 0I I c The reflected wave has: E E = E ei(!t+k1z)x^ B = 0R ei(!t+k1z)y^ R 0R R c The transmitted wave inside the conductor has: − − α − E = E ei(!t αz)e αzx^ B = (1 i)E ei(!t αz)y^ T 0T T ! − 0T For a perfect conductor ET = 0 and the wave is completely reflected 9 Boundary Conditions at Conducting Surface From H tangential: 1 1 (B0I B0R) = B0T µ0 − µ0 σ p (E E ) = (1 i)E 0 0I − 0R r2! − 0T From E tangential: E0I + E0R = E0T The reflected amplitude is: E ( σ=2! (1 i) 1) 0R = 0 − − E0I −(pσ=2! (1 i) + 1) 0 − p The sign gives a phase change π on reflection from a conductor − 10 Radiation Pressure The reflection coefficient is: E2 2! = 0R 1 2 0 2 r R E0I ≈ − σ For a good conductor σ >> 2! , 1. 0 R ! A metallic surface is a good reflector of electromagnetic waves. The reflection reverses the direction of the Poynting vector N = E H which measures energy flux × There is a radiation pressure on a conducting surface: 2< N > P = = E2 c 0 0 11 Notes: Diagrams: 12 Power Dissipation in Skin Depth The transmission coefficient is initially: E2 2! = 0T 2 0 2 r T E0I ≈ σ but this rapidly attenuates away within the skin depth Power is dissipated in the conduction currents dP = J:E = σ E 2 dτ j j Time-averaging and integrating over the skin depth: dP δ < >= p2σ! E2 = E2 c dA −2 0 0I 0 0I This result balances the radiation pressure 13 Conservation of Electromagnetic Energy The power transferred to a charge moving with velocity v is: P = F:v = qE:v In terms of current density J = Nev: P = J:Edτ ZV We can use M4 to replace J: @D J:Edτ = [E:( H) E: ]dτ ZV ZV r × − @t The first term can be rearranged using: :(E H) = H:( E) E:( H) r × r × − r × 14 Notes: Diagrams: 15 and M3 can be used to replace E: r × @D @B J:Edτ = [ :(E H) + E: + H: ]dτ ZV − ZV r × @t @t The integral over the volume dτ can be removed to give local energy conservation: @U @U J:E = [ :N + E + M ] − r @t @t Power dissipated in currents is J.E • Change in energy flux is div of Poynting vector :(E H) • r × Time variation of electric field energy density is @(D:E)=@t • Time variation of magnetic field energy density is @(B:H)=@t • 16.