Potential functions for magnetic fields

1
Potential functions for magnetic fields

• recall that in electrostatics it was sometimes
  easier to find the potential function, then get
  the electric field from the gradient
• we could either integrate over charge density to
  get V, or solve a differential equation (Poisson
  or Laplace) subject to boundary conditions
• is there a potential function for H, and is it
  easier to find??
• lets assume there is a scalar magnetic potential
  Vm that gives H from the negative gradient (just
  like in electrostatics)
• what conditions, if any, are there on Vm?

2
Potential functions for magnetic fields

• assume there is a scalar magnetic potential Vm
  that gives H from the negative gradient
• if there is current present, we already know
• so
• what is the curl of the gradient?

3
Potential functions for magnetic fields

• what is the curl of the gradient?
• in general, for any function f, the curl of the
  gradient is identically zero

4
Back to our scalar magnetic potential

• so far we have
• so it only makes sense in current free regions of
  space
• we also have from the divergence
• in current free regions Laplaces equation is
  satisfied

5
Vector magnetic potential

• turns out the scalar magnetic potential is of
  only limited usefulness
• there is another possibility
• recall for the magnetic field the divergence is
  always zero
• but for any vector field A the divergence of the
  curl of A is always zero
• so were ok if we set
• (this is the Coulomb gauge)
• A is the vector magnetic potential

6
Finding A from the current distribution

• the magnetic vector potential can be found from
• note we dont have a cross product in this!!!!
• does this make sense??
• recall
• combining

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Relation of A to H

• so far we have
• but a vector ID gives
• combining

8
A and Biot-Savart law

• so far we have
• now lets do the gradient
• with a fair amount of algebra and chain rule
  application

9
A and Biot-Savart law

• so now we have
• note that the volume integral is over the volume
  current density
• this is the same as the line integral over the
  line current we used before!
• so our definition of the vector magnetic
  potential leads directly to the Biot-Savart law!

10
Vector magnetic potential example

• lets look at a long straight wire (again)

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Vector magnetic potential straight wire

• lets make the straight wire FINITE in length,
  starting at zo, extending to z zo 2L

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Vector magnetic potential straight wire

• lets make the straight wire FINITE in length,
  starting at zo, extending to z zo 2L

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B field from the vector magnetic potential
straight wire

• FINITE length straight wire , starting at zo,
  extending to z zo 2L

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B field from the vector magnetic potential
center plane of a straight wire

• straight wire FINITE in length, starting at L,
  extending to L

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B field in the mid-plane of a straight wire

• FINITE in length straight wire, starting at L,
  extending to L

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B field mid-plane of a straight wire, more
algebra

• FINITE in length straight wire, starting at L,
  extending to L

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B field from the vector magnetic potential
straight wire, even more algebra

• so in the mid-plane of a straight wire of length
  2L

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B field from the vector magnetic potential
straight wire

• so in the mid-plane of a straight wire, length
  2L, starting at z -L and ending at z L
• now let L go to infinity
• exactly as before

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Magnetic field from two parallel wires

• one wire on the z-axis, current I in z direction
• second wire located R away, carrying the return
  current I in the z direction
• each wire extends from L to L
• lets find the vector potential in the mid-plane
• superposition on A holds

20
Magnetic field from two parallel wires

• now let the length of the wires go to infinity

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Magnetic field from two parallel wires

• lets convert to Cartesian coordinates

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Magnetic field from two parallel wires

• so in Cartesian coordinates

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Magnetic field from two parallel wires

• so in Cartesian coordinates

• lets look at x 0 (the y axis)

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Summary of magnetostatics

• for dc currents, in the absence of permanent
  magnets, summarizing everything we have so far

• now on to magnetic materials

25
B field from the vector magnetic potential
straight wire

• FINITE length straight wire , starting at zo,
  extending to z zo L

26
B field from the vector magnetic potential
straight wire

• FINITE length straight wire , starting at zo 0,
  extending to z zo L

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B field from the vector magnetic potential
straight wire

• FINITE length straight wire , starting at zo,
  extending to z zo L

28
B field from the vector magnetic potential
straight wire

• finite length straight wire, starting at zo,
  extending to z zo L

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Alternative integration for the vector magnetic
potential of a straight wire

• FINITE length straight wire , starting at zo,
  extending to z zo 2L

30
Magnetic field from two parallel wires

• now let the length of the wires go to infinity