8'3 Magnetic Vector Potential G5'4'1

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8.3 Magnetic Vector Potential (G5.4.1) To ground
this back in E M remember that we needed the
fact that to introduce the idea
of a scalar potential V in electrostatics.. So
the fact that should indicate
that we can construct a vector potential in
magnetostatics Note that is
automatically taken care of since the divergence
of a curl is always zero.
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But what about Ampères law? The term
containing is a problem if it is not
zero Remember that the electric potential had a
built-in ambiguity you can add any constant (a
function with zero gradient) to V without
altering the physically important quantity, the
electric field. We can use a similar argument
here to eliminate this term but we had better
show that we can always do this So, we can add
any function whose curl is zero (the gradient of
a scalar) to the magnetic vector potential
without affecting
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So in the end we find that But we know how
to solve this one this is three Poissons
Equations in one, one for each Cartesian
component. So assuming goes to zero at
infinity, we can just write down the solution
as and for line and surface currents
(important for magnetostatics)
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If the current does not go to zero at infinity
then we cannot use this technique to find
but there are other ways to do this which we will
explore a little later. One thing we should note
here is that is no where near as useful as V
is it is still a vector but it is still
useful in some circumstances as we will see.
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9.1 Determining Vector Potentials Example Find
the vector potential due to a long (infinite)
straight wire carrying current I. From the
vector potential determine the magnetic field.
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Example (G Ex 5.12) Find the vector potential of
an infinite solenoid with n turns per unit
length, radius R, and current I. Check that
this yields the correct magnetic field and that
the divergence of the vector potential is zero.
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Example (G Ex 5.11) A spherical shell, of radius
R, carrying a uniform surface charge s, is set
spinning at angular velocity w. Find the vector
potential in produces at point r.