fields_lec9

fields_lec9 - Key Concepts for this section 1: Lorentz...

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1: Lorentz force law, Field, Maxwell’s equation 2 : Ion Transport, Nernst-Planck equation 3: (Quasi)electrostatics, potential function, 4: Laplace’s equation , Uniqueness 5 : Debye layer, electroneutrality Goals of Part II: (1) Understand when and why electromagnetic (E and B) interaction is relevant (or not relevant) in biological systems. (2) Be able to analyze quasistatic electric fields in 2D and 3D. Key Concepts for this section
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Gel or tissue ( σ , ε ) Gel or tissue (D) Φ =V 0 Φ =0 Φ =0 Φ =0 C=0 C=0 C=0 C=C 0 Electrostatics Steady Diffusion 2 0 ∇Φ= 2 0 C ∇= e J σ =− ∇Φ r ii i JD c =− r
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222 2 0 xyz ∂Φ ∇Φ= + + = ∂∂∂ (, ,) () ()() xy zx y z Φ= Χ ϒ Ζ Assume 22 2 2 2 2 0 11 1 0 function function function of x of z of y ∂Χ ∂ϒ ∂Ζ ∇Φ=ϒΖ +ΧΖ +Χϒ = ++= Χ∂ ϒ∂ Ζ∂ 123 2 2 2 2 1 () , ( ) sin( ), cos( ) 0( ) ( , : ) xx kx x x e e x o r k x or x ax b a b constants +− = Χ= =− = + Three possibilities
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Gel or tissue ( σ , ε ) Φ =V 0 Φ =0 Φ =0 Φ =0 2 22 2 2 2 0, ( , ) ( ) ( ) 11 0 1 ()~s i n ( ) sin( ) 0 ( :integer) : n xy x y kX x k x x kL kL n n n Eigenvalue k L π ∇Φ= Φ ϒ ∂Χ ∂ϒ += Χ∂ ϒ∂ =− = = = 2 2 2 expand (x) using Fourier sine series ( ) sin (This satisfies B. C. at x=0, L) () then, ( ) 0 ( ) ~ sinh cosh ( ) sinh since ( ,0) 0 ( , ) sin
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fields_lec9 - Key Concepts for this section 1: Lorentz...

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