Laplaces Equation in Spherical polar coordinates

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Laplaces Equation in Spherical polar coordinates

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Look for a solution by separation of variables
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Rhs independent of F
Lhs depends only on F
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Rhs independent of F
Lhs depends only on F
Common value must be a constant
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Depends only on ?
Depends only on r
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Chose constant of separation to be l(l1)
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T equation
To be determined
ml is a ve integer
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T equation
Make substitution zcos?
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T equation
This equation is well known its solutions are
the associated legendre polynomials
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s
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Question
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Magnetic Vector Potentials
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• Hence

of Poissons
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• We have shown that there exists a solution A s.t.

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In cartesian coordinates
3 sets of Poissons equations
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• We may assume J goes to zero at infinity
• Then we can solve

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Electric dipole
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Multipole expansion of the vector potential

• Idea we are looking for an approximate formula
  for a localized current distribution
• We will write the potential in powers of 1/r
• Keep the highest non vanishing contribution

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quadrapole
monopole
dipole
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Vector Potential a large distance from a closed
current loop
Monopole dipole
quadrapole
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Force, Torque on current loop
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• Let us return to the circular loop of the first
  question
• The magnetic field due to a circular loop at a
  point on the z-axis is

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• Field lines due to
• An electric dipole
• A current loop
• A bar magnet (dashed lines are inside the magnet)