Download * Magnetic Scalar Potential * Magnetic Vector Potential

PPT No. 19
* Magnetic Scalar Potential
* Magnetic Vector Potential
Magnetic Potentials
The Magnetic Potential is
a method of representing the Magnetic field
by using a quantity called Potential
instead of the actual B vector field.
Magnetic Potentials
Magnetic field can be related to a potential by two methods
which give rise to two possible types of magnetic potentials
used in different situations:
1. Magnetic Scalar Potential
2. Magnetic Vector Potential
A) Magnetic Scalar Potential
In Electrostatics,
electric field E is derivable from the electric potential V.
V is a scalar quantity and easier to handle than
E which is a vector quantity.
In Magnetostatics,
the quantity Magnetic scalar potential can be obtained
using analogues relation
A) Magnetic Scalar Potential
In regions of space in the absence of currents,
the current density j =0
=0
B is derivable from the gradient of a potential
Therefore B can be expressed as
the gradient of a scalar quantity φm
B = - ∇φm
φm is called as the Magnetic scalar potential.
A) Magnetic Scalar Potential
The presence of a magnetic moment m creates
a magnetic field B
which is the gradient of some scalar field φm .
The divergence of the magnetic field B is zero,
∇.B = 0
By definition, the divergence of the gradient of
the scalar field is also zero,
- ∇.∇φm = 0
or
∇2 φm = 0.
The operator ∇2 is called the Laplacian and
∇2 φm = 0 is the Laplace’s equation.
A) Magnetic Scalar Potential
∇2 φm = 0
Laplace’s equation is valid
only outside the magnetic sources and
away from currents.
Magnetic field can be calculated
from the magnetic scalar potential
using solutions of Laplace’s equation.
B) Magnetic Vector Potential
The magnetic scalar potential is useful
only in the region of space away from free currents.
If J=0, then only magnetic flux density can be computed
from the magnetic scalar potential
The potential function which overcomes this limitation
and is useful to compute B
in region where J is present is
.
Magnetic Vector Potential
B) Magnetic Vector Potential
Magnetic fields are generated by
steady (time-independent) currents &
satisfy Gauss’ Law
Since the divergence of a curl is zero,
B can be written as the curl of a vector A as
B) Magnetic Vector Potential
Any solenoidal vector field (e.g. B) in physics can
always be written as
the curl of some other vector field (A).
The quantity A is known as
the Magnetic Vector Potential.
B) Magnetic Vector Potential
{However, magnetic vector potential
is not directly associated with work
the way
that scalar potential (e.g. Electric potential V)
is associated with work}
Work done against the electric field E
is stored as electric potential energy U
given in terms of
electric dipole moment p and E as
B) Magnetic Vector Potential
The vector potential is defined
to be consistent with Ampere’s Circuital Law and
It can be expressed in terms of either current i or
current density j
(i.e. the sources of magnetic field) as follows
B) Magnetic Vector Potential
However, A is Not uniquely defined by the above equation.
Any function whose curl is zero, can be added to A,
then the result would still be the same field B. e.g.
If ∇ψ, the Gradient of a scalar ψ is added to A
∇ x (A + ∇ψ )=∇ x A + ∇ x ∇ψ = ∇ x A = B
B) Magnetic Vector Potential
To make A more specific/ unique,
additional condition needs to be imposed on A.
In Magnetostatics a convenient condition which makes
calculations easier can be specified as ∇. A = 0
(In Electrodynamics, this condition cannot be imposed)
B) Magnetic Vector Potential
The set of equations which
uniquely define the vector potential A and
also satisfy the fundamental equation of Gauss’ Law
∇. B = 0
{the magnetic field is divergence-free},
are as follows
B) Magnetic Vector Potential
From Ampere’s law
Therefore the equation
can be written as
This equation is similar to Poisson's equation,
the only difference is that A is a vector.
B) Magnetic Vector Potential
Each component (e.g. along x, y, z axes) of A
must satisfy the differential equation of the type
A unique solution to the above Poisson's equation
can be found
(By combining the solutions for components on x, y, z).
It specifies the magnetic vector potential A
generated by steady currents.
B) Magnetic Vector Potential
First A is determined using Poisson's equation then
it is substituted in the equation
Thus the field B produced by a steady current
can be computed.
Gauge Transformation
According to Helmholtz's theorem
a vector field is fully specified by
its divergence and its curl.
The curl of the vector potential A
gives the magnetic field B via Eq.
However, the divergence of A
has no physical significance
can be chosen freely as desired
B) Magnetic Vector Potential
According to the equation
the magnetic field is invariant under the transformation
In other words,
the vector potential is undetermined
to the gradient of a scalar field
can be chosen as desired
B) Magnetic Vector Potential
The electric scalar potential is undetermined
to an arbitrary additive constant, since the transformation
leaves the electric field invariant in Equation
The transformations
and
are examples of gauge transformations in Mathematics.
B) Magnetic Vector Potential
In electromagnetic theory,
several "gauges" have been used to advantage
depending on the specific types of calculations
The choice of a particular function ψ or
a particular constant c is referred to
as a choice of the gauge.
B) Magnetic Vector Potential
The gauge can be fixed as desired.
Usually it is chosen to make equations simplest possible.
It is convenient to choose gauge for the scalar potential Ф
such that Ф → 0 at infinity.
The gauge for A is chosen such that
This particular choice
is known as the Coulomb gauge