Introduction to Mathematical Physics/Electromagnetism/Electromagnetic induction

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Introduction[edit]

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 Preliminaries[edit]

Loops, multi-loops, and divergence-free vector fields[edit]

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[edit]

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 Inductance[edit]

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 Inductance[edit]

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

Linearity of Mutual Inductance[edit]

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 Inductance[edit]

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 Inductance[edit]

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.