Introduction to Mathematical Physics/Electromagnetism/Electromagnetic induction

IntroductionEdit

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.


Mathematical PreliminariesEdit

Loops, multi-loops, and divergence-free vector fieldsEdit

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 fieldsEdit

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.

Definition of Mutual InductanceEdit

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  

Self InductanceEdit

When  , the inductance   is referred to as the "self inductance".

Linearity of Mutual InductanceEdit

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  .

Symmetry of Mutual InductanceEdit

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.

Calculating the Mutual InductanceEdit

Approach #1 (use the vector potential)Edit

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:  .

Approach #2 (use linearity and loop dipoles)Edit

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.

  1. Griffiths, D. J., Introduction to Electrodynamics, 3rd edition, Prentice Hall, 1999.