Coulomb

1
Coulombs Law and Fields In Vacuum

• Like charges repel, unlike charges attract
• Force is directly along a line joining the two
  charges

q1
q2
r
2
Electric Fields

• An electric Field is a local vector quantity
  incorporating the ability to exert a force at a
  distance on a charge

From point charge q
test charge q
3
Potential Energy of charges
charge q

• If we have a field, then we have a potential
  energy

Electric Field E

• Factor out the charge then you have
  electrostatic potential V

4
Electrostatic Potential and Field

• Does everyone know what the del operator is?

• The potential becomes the fundamental quantity
  electrostatics.

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Gausss Law
charge q
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Result of Stokes Thm
Reminder
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Result of Stokes Thm
Poissons Equation
So if we know the charge distribution, we can
calculate the electrostatic potential! and hence
forces and energies on charges!
8
Dielectric constants
Arise from polarization orientation and
induction of dipole moments, which weaken the
electric field between two charges
Dielectric constants allow one to average over
details
Both the mobility and presence of dipole moments
both affect dielectric constants
9
Dielectric constants
Arise from polarization orientation and
induction of dipole moments, which weaken the
electric field between two charges
10
Dielectric constants
Arise from polarization orientation and
induction of dipole moments, which weaken the
electric field between two charges
Dielectric constants allow one to average over
details
Both the mobility and presence of dipole moments
both affect dielectric constants
11
Heterogenous Dielectric constants
If the system of interest is heterogenous, than
the dielectric constant can be heterogenous
Given what we know about protein structure and
dynamics should a protein have a single
dielectric constant ? Why or why not?
12
Heterogenous Dielectric constants Special case
Two-dielectric system
ew80
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The previous equations simplify 1) Special
symmetries give rise to nice analytic
solutions 2) For practical, biomolecular
calculations, computers are needed. Why? 3) For
practical calcs, the utility of two-dielectrics
systems are A)Fewer parameters B) One can
define a reaction field
13
Two Dielectric Systmes
The potential can be divided into two pieces
Potential from polarization of the medium
Potential from the charges inside



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14
Finite-Differences
In order to solve this set of equations on a
computer one has to use finite-difference
methods. Derived from finite-approximations to
deriviatives.
Definition of derivative from elementary calculus
Simplest Finite difference approximation to a
derivative is
15
Finite-Difference
Iterate to obtain a simple expression for a
second derivative
For a vector derivative
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Finite-Difference
To improve on these schemes, use Taylor
expansions
And finite difference approxs
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Finite-Difference PB