Newton’s law m (∂v/∂t) = q(E + v x µH) + f other

µ

(1)

(2)

(3)

(4)

(5)

(6)

(7)

f elec

v

(single charged particle)

Constitutive Laws for Linear, Isotropic Media

D = εE = εoE + P

B = µH

J = σE

Complete Description of Electrodynamics1

© Garland Science. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/.Source: Grodzinsky, Alan. Field, Forces and Flows in Biological Systems. Garland Science, 2011. [Preview with Google Books]

Newton’s law: m (∂v/∂t) = q(E + v x µH) + f other

µ

(1)

(2)

(5)

(6)

(7)

v

(single charged particle)

Complete Description of Electrodynamics

≈ 0 either H = 0, or• ∂/∂t ≈ small• low enough freq• λ >> Lchar

J = σE +…

Constitutive Law

2

© Garland Science. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. Source: Grodzinsky, Alan. Field, Forces and Flows in Biological Systems. Garland Science, 2011. [Preview with Google Books]

J = σEOhm's Law:

1789-1854

Mathematician and experimentalist

Current Flow in conductors

(empirical)

3

+Vo-

Electro-Statics:

x = 0 x = Ly = 0

y = d

4

+Vo-

But: Electrolysis Reactions at Electrodes

Metal Electrodes

Really: J = σE + diffusion + convection

E

J = σE

Gas bubbles* *

DEMO

5

+Vo-

ElectroStatics: ∇•J = -(∂ρe/∂t) ≡ 0

x = 0 x = L

E = -∇Φ = ix (Vo/L)^

Φ = Vo (1 – x/L) V=0

∇•J = 0 = ∇•σE = σ[∇•(-∇Φ)] = 0 → ∇2Φ = 0 Laplace

∂2Φ∂x2 = 0

(conductivity, σ)(permittivity, ε)

(J = σE )

6

Table B.7 page 297

Solutions of Laplace’s Eq.

in 2 - dimen’s

∇2Φ = 0

7

PSet 4, P3: Gradient Gel Electrophoresis

∇2Φ = 0 ∇2Φ = 0

Splice solutions together via appropriate Boundary Conditions

(J = σE) (J = σE)

Φ = 0 Φ = + Vo

X = L/2 X = L

y = 0

y = d

8

LAW Boundary Cond.

Gauss

Faraday

⇒ E = -∇Φ

(4) J = σE (+ other)

Cons. of Charge

EQS subset of Maxwell's Eqns

0 static

*

*(constitutive law)

9

+Vo-

Electro-Statics:

x = 0 x = L

ρe (coul) = Σ zi F(coul) ci (mol) m3 mol m3i

ρe = 0 in “bulk”

∇•εE = ρe = 0 → ∇2Φ = 0 Laplace

10

+Vo-

But is ρe zero everywhere?

x = 0 x = L

ρe = 0 in “bulk”

J = σE (in bulk)

∇•εE = ρe = 0 → ∇2Φ = 0 Laplace

(ρe ≠ 0)

11

Molecular Interactions

Force

Force

molecular Electrostatic Interactions

molecular Electrostatic Interactions

Poisson-Boltzmann: Molecular Electrostatic Interactions

COO + H COOH +→←Na+

Cl

12

PSet 4, P2

Poisson’s Eqn

“Poisson-Boltzmann Eqn”(linearized)

….Find Φ(x)

Φ(x)

x

+

+

+

+

Φo

at x=0BC:

0.1M NaCl

Cl_

Na+

Φ(x)(ρe ≠ 0)

Φ(x)→0

σdSurface charge

13

Newton’s law: m (∂v/∂t) = q(E + v x µH) + f other

µ

(1)

(2)

(5)

(6)

(7)

v

(single charged particle)

Complete Description of Electrodynamics

≈ 0 either H = 0, or• ∂/∂t ≈ small• low enough freq• λ >> Lchar

J = σE +…

Constitutive Law

14

© Garland Science. All rights reserved. This content is excluded from our Creative Commons license. For more information, see http://ocw.mit.edu/help/faq-fair-use/. Source: Grodzinsky, Alan. Field, Forces and Flows in Biological Systems. Garland Science, 2011. [Preview with Google Books]

FFF: Complete Description of Coupled Transport and Biomolecular Interactions

"E.Q.S."

NavierStokes

Diffusion-Reaction

C E

MCurrent Density J

15

MIT OpenCourseWarehttp://ocw.mit.edu

20.430J / 2.795J / 6.561J / 10.539J Fields, Forces, and Flows in Biological SystemsFall 2015

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.