EE204B_Lesson_10_E-field_in_Material-2_

EE204B_Lesson_10_E-field_in_Material-2_ - EE204B EE204B...

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Unformatted text preview: EE204B EE204B Electromagnetics Electromagnetics Lesson Lesson 10 Electric Fields in Material Space Continuity Eq., Boundary Coditions Instructor: Prof. Yong H. Won Lesson 10 Topics Electric Field in Material Space (Chap. 5) • Continuity Equation and Relaxation Time Eq Ti • Boundary Conditions • Dielectric-Dielectric Interface • Conductor-Dielectric Interface • Conductor-Free Space Interface EE204B Electromagnetics Prof. Yong H. Won Spring 2010 2 Lesson 10 Conductors or Dielectrics EE204B Electromagnetics Prof. Yong H. Won Spring 2010 3 Lesson 10 Conductors or Dielectrics EE204B Electromagnetics Prof. Yong H. Won Spring 2010 4 Lesson 10 Continuity Equation Charge conservation principle Rate of charge within a volume = Net flow of current through surface enclosing volume. ΔQ ↓ Net outflow of current ΔQ ↑ Net outflow of current Mathematically, I out = ∫ J ⋅ dS = −dQin ∂ρ d = − ∫ ρv dv = − ∫ v dv v ∂t dt dt v Using the divergence theorem Using the divergence theorem Finally rr ∂ρv ∫v ∇ ⋅ Jdv = −∫v ∂t dv ∂ρv r r ρ or + ∇ ⋅ J = 0 Continuity Equation ∂t Prof. Yong H. Won Spring 2010 5 EE204B Electromagnetics , ∫ s rr r J ⋅ dS = ∫ ∇ ⋅ Jdv v Lesson 10 Dielectric Relaxation Time Consider now putting charge in materials (conductor or dielectric) di ΔQ ≠ 0 r r r ρv Using Ohm’s law: J = σE and Gauss’s law: ∇ ⋅ E = r σρv ε ∂ρv Using Continuity Eq. ∇ ⋅ σE = =− ε ∂t ∂ρv σ →∴ + ρv = 0 : ∂t ε 1st order D.E. Solve the equation the equation d ρv σ = − dt → ρv ε σt ln ρv = − + ln ρvo ε where Tr = ε/σ = relaxation/ rearrangement time = Time taken for charge in material to drop to e-1 charge in material to drop to (36.8%) of its initial value Spring 2010 6 ∴ ρv = ρvo e−σt / ε ≡ ρvo e−t / Tr EE204B Electromagnetics Prof. Yong H. Won Lesson 10 Dielectric Relaxation Time Conductors: Short Tr Sh Charges redistribute fast e.g. Copper σ = 5.8×107 S/m, εr = 1.0 ε r εo 10−9 1 Tr = = 1⋅ ⋅ = 1.53 ×10−19 [sec] 36π 5.8 ×107 σ Dielectrics: Longer Tr Slow rearrangement e.g. Fused quartz σ =10-17 S/m, εr = 5.0 ε r εo 10−9 1 Tr = = 5⋅ ⋅ −17 = 51.2 [days] 36π 10 σ Good dielectrics capability of storing charges Use in capacitors Prof. Yong H. Won Spring 2010 7 EE204B Electromagnetics Lesson 10 Electric Boundary Conditions EE204B Electromagnetics Prof. Yong H. Won Spring 2010 8 Lesson 10 Electric Boundary Conditions EE204B Electromagnetics Prof. Yong H. Won Spring 2010 9 Lesson 10 Electric Boundary Conditions EE204B Electromagnetics Prof. Yong H. Won Spring 2010 10 Lesson 10 Electric Boundary Conditions EE204B Electromagnetics Prof. Yong H. Won Spring 2010 11 Lesson 10 Electric Boundary Conditions z Example 10-1 εr = 2.5 in x < 0 and εr = 1 in x > 0 r r r r If in If in x < 0, D1 = 12ax − 10a y + 4az nC/m2 nC/m r r Find D2 which is D in x > 0 y x εr=2.5 εr=1 Solution D2 x = D1x = 12 E2 y = E1 y = D’s normal component is continuous when ρs = 0, Here x is normal to the interface, y and z are tangential to it. −10 , 2.5εo E2 z = E1z = 2 4 2.5εo 4 D2 z = εo E2 z = = 1.6 2.5 Spring 2010 12 ∴ D2 y = εo E2 y = −4 nC/m , r r r r ∴ D2 = 12ax − 4a y + 1.6az nC/m2 EE204B Electromagnetics Prof. Yong H. Won Lesson 10 Electric Boundary Conditions EE204B Electromagnetics Prof. Yong H. Won Spring 2010 13 Lesson 10 Electric Boundary Conditions 10-2 [Text Example 5.9] EE204B Electromagnetics Prof. Yong H. Won Spring 2010 14 Lesson 10 Electric Boundary Conditions EE204B Electromagnetics Prof. Yong H. Won Spring 2010 15 Lesson 10 Electric Boundary Conditions Example 10-3 [Text Example 5.10] A perfect conductor is located at y < 0. perfect conductor is located at A dielectric medium (εr = 2) is at y > 0. If there is a surface charge of 2 nC/m2 on the conductor, there is surface charge of nC the conductor determine E and D at (a) (3 (a) A (3, -2, 2) 2) (b) B (-4, 1, 5) EE204B Electromagnetics Prof. Yong H. Won Spring 2010 16 ...
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This note was uploaded on 05/14/2010 for the course EE EE204 taught by Professor Parkkyungsoo during the Spring '10 term at Korea University.

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