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.
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