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 ∇ C ∇ ∂B E = ∇ × − ∂t but M4 in conductors includes a free current density JC = σE: ∂D H = + JC ∇ × ∂t ∂E B = µ   + µ σE ∇ × 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 ∇ × ∇ × −∂t ∇ × ∇ ∇ − ∇ using M1 with the assumption that the free charge density is zero:

ρ =   ( .E) = 0 ∇ c r 0∇ ∇ Subtituting for B from M4 leads to a modified wave equation: ∂ ∂2E ∂E 2E = ( B) = µ   + µ σ ∇ ∂t ∇ × 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:

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4 Plane Waves in Conductors

The solution can be written as an attenuated plane wave:

i(ωt−βz) −αz Ex = E0e e

Substituting this back into the modified wave equation: ( iβ α)2E = µ   (iω)2E + µ σ(iω)E − − 0 0 r 0 0 0 0 Equating the real and imaginary parts: β2 + α2 = µ   ω2 2βα = µ σω − − 0 r 0 0 For a good conductor with σ   ω:  r 0 µ σω α = β = 0 r 2 For a perfect conductor σ there are no waves and E = 0 → ∞

5 Skin Depth

The attenuation length in a conductor is known as the skin depth: 1 2 δ = = α rµ0σω

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(ωt αz)e αz ∂z − x − 0 where we neglect the term in the wave equation proportional to r

σE − − − − − H = 0 ei(ωt αz)e αz = H ei(ωt 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 ω: H σ 0 = E0 rµ0ω In a good conductor H E and they are no longer related by c! 0  0

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8 Reflection at a Conducting Surface

We just consider the case of normal incidence The incident wave has:

− E − E = E ei(ωt k1z)xˆ B = 0I ei(ωt k1z)yˆ I 0I I c The reflected wave has: E E = E ei(ωt+k1z)xˆ B = 0R ei(ωt+k1z)yˆ R 0R R c The transmitted wave inside the conductor has:

− − α − E = E ei(ωt αz)e αzxˆ B = (1 i)E ei(ωt αz)yˆ T 0T T ω − 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 √ (E E ) = (1 i)E 0 0I − 0R r2ω − 0T From E tangential:

E0I + E0R = E0T The reflected amplitude is: E ( σ/2ω (1 i) 1) 0R = 0 − − E0I −(pσ/2ω (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 2ω = 0R 1 2 0 2 r R E0I ≈ − σ For a good conductor σ >> 2ω , 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:

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12 Power Dissipation in Skin Depth

The transmission coefficient is initially:

E2 2ω = 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 | |

Time-averaging and integrating over the skin depth: dP δ < >= √2σω 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 ∇ × − ∂t The first term can be rearranged using:

.(E H) = H.( E) E.( H) ∇ × ∇ × − ∇ ×

14 Notes:

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15 and M3 can be used to replace E: ∇ × ∂D ∂B J.Edτ = [ .(E H) + E. + H. ]dτ ZV − ZV ∇ × ∂t ∂t The integral over the volume dτ can be removed to give local energy conservation: ∂U ∂U J.E = [ .N + E + M ] − ∇ ∂t ∂t

Power dissipated in currents is J.E • Change in energy flux is div of Poynting vector .(E H) • ∇ × Time variation of electric field energy density is ∂(D.E)/∂t • Time variation of magnetic field energy density is ∂(B.H)/∂t •

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