Phys 2214 hw7 Solutions

Phys 2214 hw7 Solutions - Phys 2214 Homework#7 Solutions...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Phys 2214 Homework #7 Solutions November 1, 2011 1. EM Waves in Conductors. The current density is ~ J = ~ E/ρ , where ρ is the electrical resistivity. Ampere’s law in differential form is ∇ × ~ B = μ ρ ~ E + μ ∂ ~ E ∂t . If ρ is small (a good conductor), then the first term on the right hand side will dominate and we can neglect the second term. (a) Faraday’s law is ∇ × ~ E =- ∂ ~ B/∂t . Taking the curl of this gives ∇ × ( ∇ × ~ E ) =-∇ × ∂ ~ B ∂t =- ∂ ∂t ∇ × ~ B , where the spatial and temporal derivatives have been interchanged. Replace the left hand side using the vector identity ∇× ( ∇× ~ E ) = ∇ ( ∇ · ~ E )- ∇ 2 ~ E . Then ∇ ( ∇ · ~ E )- ∇ 2 ~ E =- ∂ ∂t ( ∇ × ~ B ) . We know that ∇· ~ E = 0, from Gauss’ law with no charge density, and ∇ × ~ B = ( μ/ρ ) ~ E from Ampere’s law in a conductor. Then ∇ 2 ~ E = μ ρ ∂ ~ E ∂t . 1 (b) Let ~ E ( z,t ) = E x e i ( γz- ωt ) ˆ i . Sugstituting into the above wave equation gives ( iγ ) 2 ~ E = ( μ/ρ )(- iω ) ~ E . Then- γ 2 =- iμω/ρ , or γ = q iμω/ρ . Using √ i = (1+ i ) / √ 2, we get γ = q μω/ 2 ρ (1+ i ) ≡ γ (1 + i ). (c) Substituting this equation for γ into the plane wave solution gives ~ E = E x e i [ γ (1+ i ) z- ωt ] ˆ i = E x e- γ z e i ( γ z- ωt ) ˆ i , which is a plane traveling wave whose amplitude decays with dis- tance z into the conductor. (d) The electric field decays to 1 /e of its initial value when γ δ = 1, or δ = q 2 ρ/μω (the skin depth). (e) Assume μ = μ = 4 π × 10- 7 Tm/A . i. For Cu, ρ = 1 . 7 × 10- 8 Ω · m . For f = 60 Hz , δ = 8 . 47 × 10- 3 m = 8 . 47 mm . For f = 840 MHz , δ = 2 . 3 × 10- 6 m = 2 . 3 μm ....
View Full Document

{[ snackBarMessage ]}

Page1 / 8

Phys 2214 hw7 Solutions - Phys 2214 Homework#7 Solutions...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online