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F16 PH333 3.4.1-3.4.4

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Richard Datwyler

on 29 October 2018

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Transcript of F16 PH333 3.4.1-3.4.4

Multipole Expansion
3.4.1 Potentials at great distances
3.4.2 Monopole and dipole terms
3.4.3 The origin...
3.4.4 E field from dipole
Big picture
When far away from charge distrabutions they often 'look' and behave like a point charge, but what about higher order terms.
Multipole expansion: shows these higher terms in powers of 1/r. Take them as far as you want
Nice to see and understand which will be the first term of significance in the expansion.
Dipole
d
r
-q
+q
point: potential of dipole drops off as 1/r^2
Naming and relationships
Monopole
Dipole
Quadrupole
Octopole
.....
1/r
1/r^2
1/r^3
1/r^4
.....
Posit. new expansion for charge distributions in powers of 1/r
Fun with "R" generating function
This then is the multipole expansion
monopole
dipole
quadrupole
This is an approximation go as far as you want.
Monopole
This is the sum of the total charge
giving the potential as
note a point charge is a monopole
if the net charge is zero, this term vanishes
Dipole
recall
define
Origin of multipole expansions
All I want to say here is the coordinates and the distance from the origin matters in multipole expansions
Only exception to that is the dipole of net neutral charge configurations.
E field
-2q
-2q
3q
q
z
y
x
4 charges are shown, each a distance a from the origin. Find a simple approx. formula for the potential far away.
a
First : monopole, dipole, quadpole.....
what is the leading term?
Find the dipole moment!
sort out the dot product
final answer
z
y
x
a
-q
-q
q
Find E field from two lowest order multipole expansions
First question what are they?
dipole
monopole
sort out the dipole moment and dot product and final answer V
Now the E field
"When we do the multipole expansion do we take our r from the center of the charge distribution?"
"When the total charge of a system is zero, then the dipole moment is independent of the choice of origin. Why is that? Can we do an example? Also, the math on calculating the moment is a little confusing to me still. "
"What do all of the r', rho' tau' mean in the integral?"
"I can tell that we are starting to work with more complex systems but I get lost when the pictures end and the math starts. what does the multiple expansion specifically refer to?"
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