Electromagnetic induction refers to the induction of an electric motive force (emf) in a closed loop
via Faraday's law from the magnetic field generated by current in a closed loop
.
The two laws involved in electromagnetic induction are:
Ampere's Law (static version):
Faraday's Law:
where
and
are the electric and magnetic fields respectively,
is the current density and
is the magnetic permeability.
The relationship between paths, loops, and divergence free vector fields is an important mathematical preliminary that merits a brief introduction.
Given any oriented path
,
can be characterized by a vector field
.
for all positions
. For all positions
,
is infinite in the direction of
in a manner similar to the Dirac delta function. The integral property that must be satisfied by
is that for any oriented surface
, if
passes through
in the preferred direction a net total of
times, then
(
is a vector that denotes an infinitesimal oriented surface segment)
(
passing through
in the reverse direction decreases
by 1.)
Given any vector field
,
(
is a vector that denotes an infinitesimal oriented path segment, and
is an infinitesimal volume segment)
It is easy to verify that if
is a closed loop, then
Given any sequence of closed loops
, these loops can be added in a linear fashion to get a "multi-loop" denoted by the vector field
. The multi-loop is denoted by:
.
Most importantly, given any divergence-free vector field
that decreases faster than
as
, then there exists a family
of closed loops where
is an arbitrary continuous indexing parameter such that
. In simpler terms, any divergence free vector field can be expressed as a linear combination of closed loops.
Surfaces, multi-surfaces, and irrotational vector fields
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The relationship between surfaces, closed surfaces, and irrotational vector fields is also an important mathematical preliminary that merits a brief introduction.
Given any oriented surface
,
can be characterized by a vector field
.
for all positions
. For all positions
,
is infinite in the direction of the outwards normal direction to
in a manner similar to the Dirac delta function. The integral property that must be satisfied by
is that for any oriented path
, if
passes through
in the preferred direction a net total of
times, then
(
passing through
in the reverse direction decreases
by 1.)
Given any vector field
,
It is easy to verify that if
is a closed surface, then
is irrotational.
Given any sequence of surfaces
, these surfaces can be added in a linear fashion to get a "multi-surface" denoted by the vector field
. The multi-surface is denoted by:
.
Most importantly, given any irrotational vector field
that decreases faster than
as
, then there exists a family
of closed surfaces where
is an arbitrary continuous indexing parameter such that
. In simpler terms, any irrotational vector field can be expressed as a linear combination of closed surfaces.
Given an oriented surface
with a counter-clockwise oriented boundary
, it is then the case that
. Given any vector field
that denotes a multi-surface, then
is a vector field that denotes the counter-clockwise oriented boundary of the multi-surface denoted by
. This property is important as it enables a magnetic field to denote a multi-surface interior for the closed loop of current that generates it.
Let
and
be two oriented closed loops, and let
and
be oriented surfaces whose counter-clockwise boundaries are respectively
and
.
Given a current of
flowing around
, let
be the magnetic field induced via Ampere's law. Note that
. The magnetic flux through surface
is
where
is the vector representation of an infinitesimal surface element of
.
Note that also,
. This constant of proportionality,
, is the mutual electromagnetic induction from
to
.
The mutual electromagnetic induction from
to
will be denoted with
When
, the inductance
is referred to as the "self inductance".
Given loops
,
, and
, it is relatively simple to demonstrate that
and
.
Let
,
, and
be the magnetic fields generated when a current of
flows through
,
, or
respectively.
The magnetic field generated by
and
together is
due to the linearity of Maxwell's equations. This leads to
.
The flux through
is the sum of the flux through
and
separately. This leads to
.
It is the case that given loops
and
, that
. This symmetry, while apparent from explicit formulas for the mutual inductance, is far from obvious however. To make this fact more intuitive, the magnetic fields that are generated by
and
will be interpreted as multi-surfaces whose boundaries are respectively
and
.
Let there exist a current of
in loop
, and let
denote the resultant magnetic field. Ampere's law requires that
, and therefore
is a multi-surface whose boundary is
. Since
, let
.
Given a divergence free vector field
, the flux of
through
is:
The final equality holds due to the fact that
is divergence free and that
and
are multi-surfaces with a common boundary of
.
is divergence free. The flux of
through
is:
Therefore:
from which the symmetry
is now apparent.
Gauss' law of magnetism requires that
. This makes possible a "vector potential" for
: a vector field
which satisfies
. The condition
can also be enforced.
Using the vector identity:
For any vector field
:
Ampere's law becomes:
is an instance of Poisson's equation which has the solution:
It can be checked that for this solution, since
, that
.
The vector potential generated by a current of
flowing through closed loop
is:
The magnetic field generated by a current of
flowing through closed loop
is:
. The flux through surface
(which is counter-clockwise bounded by
), is
via Stoke's theorem.
so the mutual inductance is:
This equation is known as "Neumann formula" [1].
It can also be seen from this expression that the mutual inductance is symmetric:
.
Given any closed loop
, let
be an oriented surface that has
as its counterclockwise boundary. For each infinitesimal area vector element
of
, let the infinitesimal
be an infinitesimal closed loop that is the counterclockwise boundary of
. It is then the case that
.
The linearity of the mutual inductance gives:
In other words, the mutual inductance between two large loops can be expressed as the sum of mutual inductances between several mini loops.
Given an area vector
, and a current
that flows around the boundary of
in a counterclockwise manner, then the magnetic dipole (vector) formed is
. If the area shrinks, then the current increases proportionally if the magnetic dipole is to remain constant.
Given a magnetic dipole
with an infinitesimal area at position
, the magnetic field produced by
is:
Let
and
be area vectors of the interiors of two infinitesimal loops, with the second loop displaced from the first by
. Let a current
flow around the boundary of
in a counter clockwise manner forming the dipole
. The flux of the magnetic field generated by
through
is:
Therefore if
and
are the counter clockwise boundaries of
and
:
Returning to computing the mutual inductance between
and
gives:
This formula is centered around surface integrals as opposed to loop integrals.
- ↑ Griffiths, D. J., Introduction to Electrodynamics, 3rd edition, Prentice Hall, 1999.