329fall08hw7sol - ECE-329 Fall 2008 Homework 7 — Solution...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ECE-329 Fall 2008 Homework 7 — Solution October 15, 2008 1. In HW6, we found that the charge density in a conductor with conductivity σ and permittivity o is ρ ( r ,t ) = cos(100 x ) e- σ o t C / m 3 . a) Using the continuity equation ∇ · J + ∂ρ ∂t = 0 and noting that J must have only a component along x-direction (i.e., J = J x ˆ x ), we can write ∂J x ∂x =- ∂ρ ∂t . Solving this equation we obtain J = σ 100 o sin(100 x ) e- σ o t ˆ x A m 2 . The associated electric field is E = 1 σ J = 1 100 o sin(100 x ) e- σ o t ˆ x V m . b) The energy dissipation rate per volume in the conductor is simply J · E = σ 10 4 2 o sin 2 (100 x ) e- 2 σ o t W m 3 . 2. For the specified charge distribution, let us compute D , E , and P in all regions, as well as, the surface charge densities on each boundary. a) Region 1 ( r ≤ a ) is occupied by a conductor with σ = 10 6 S / m , therefore, we can directly write D 1 = C m 2 , E 1 = V m , P 1 = C m 2 . In steady-state, charges can accumulate only on the surface of conducting materials. Since this material holds a net charge Q 1 = 1 C , the surface charge density on the sphere of radius r = a is ρ S | r = a = Q 1 Area = 1 4 πa 2 C m 2 . b) Region 2 ( a < r < b ) is occupied by a dielectric with 2 = 5 o . Applying Gauss’ law ¸ D · d S = Q enc and considering D = D r ˆ r , we find D 2 = 1 4 πr 2 ˆ r C m 2 for a < r < b. In addition, we get E 2 = 1 2 D 2 = 1 20 π o r 2 ˆ r V m , and P 2 = D 2- o E 2 = 1 4 πr 2- 1 20 πr 2 ˆ r = 1 5 πr 2 ˆ r C m 2 ....
View Full Document

This note was uploaded on 04/14/2009 for the course ECE 329 taught by Professor Franke during the Spring '08 term at University of Illinois at Urbana–Champaign.

Page1 / 5

329fall08hw7sol - ECE-329 Fall 2008 Homework 7 — Solution...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online