# Introduction to Mathematical Physics/Electromagnetism/Electromagnetic induction

## Introduction

Electromagnetic induction refers to the induction of an electric motive force (emf) in a closed loop ${\displaystyle C_{2}}$  via Faraday's law from the magnetic field generated by current in a closed loop ${\displaystyle C_{1}}$ .

The two laws involved in electromagnetic induction are:

Ampere's Law (static version): ${\displaystyle \nabla \times B=\mu _{0}j}$

Faraday's Law: ${\displaystyle \nabla \times E=-{\frac {\partial B}{\partial t}}}$

where ${\displaystyle E}$  and ${\displaystyle B}$  are the electric and magnetic fields respectively, ${\displaystyle j}$  is the current density and ${\displaystyle \mu _{0}}$  is the magnetic permeability.

## Mathematical Preliminaries

### Loops, multi-loops, and divergence-free vector fields

The relationship between paths, loops, and divergence free vector fields is an important mathematical preliminary that merits a brief introduction.

Given any oriented path ${\displaystyle C}$ , ${\displaystyle C}$  can be characterized by a vector field ${\displaystyle \delta (r;C)}$ . ${\displaystyle \delta (r;C)=0}$  for all positions ${\displaystyle r\notin C}$ . For all positions ${\displaystyle r\in C}$ , ${\displaystyle \delta (r;C)}$  is infinite in the direction of ${\displaystyle C}$  in a manner similar to the Dirac delta function. The integral property that must be satisfied by ${\displaystyle \delta (r;C)}$  is that for any oriented surface ${\displaystyle \sigma }$ , if ${\displaystyle C}$  passes through ${\displaystyle \sigma }$  in the preferred direction a net total of ${\displaystyle N}$  times, then

${\displaystyle \iint _{r\in \sigma }\delta (r;C)\bullet dA=N}$  (${\displaystyle dA}$  is a vector that denotes an infinitesimal oriented surface segment)

(${\displaystyle C}$  passing through ${\displaystyle \sigma }$  in the reverse direction decreases ${\displaystyle N}$  by 1.)

Given any vector field ${\displaystyle F(r)}$ , ${\displaystyle \int _{r\in C}F(r)\bullet dr=\iiint _{r\in \mathbb {R} ^{3}}(F(r)\bullet \delta (r;C))d\tau }$  (${\displaystyle dr}$  is a vector that denotes an infinitesimal oriented path segment, and ${\displaystyle d\tau }$  is an infinitesimal volume segment)

It is easy to verify that if ${\displaystyle C}$  is a closed loop, then ${\displaystyle \nabla \bullet \delta (r;C)=0}$

Given any sequence of closed loops ${\displaystyle C_{1},C_{2},\dots ,C_{k}}$ , these loops can be added in a linear fashion to get a "multi-loop" denoted by the vector field ${\displaystyle \delta (r;C_{1})+\delta (r;C_{2})+\dots +\delta (r;C_{k})}$ . The multi-loop is denoted by: ${\displaystyle C_{1}+C_{2}+\dots +C_{k}}$ .

Most importantly, given any divergence-free vector field ${\displaystyle F}$  that decreases faster than ${\displaystyle o(1/|r|^{2})}$  as ${\displaystyle |r|\rightarrow +\infty }$ , then there exists a family ${\displaystyle C[\xi ]}$  of closed loops where ${\displaystyle \xi \in D_{C}}$  is an arbitrary continuous indexing parameter such that ${\displaystyle F(r)=\iint _{\xi \in D_{C}}\delta (r;C[\xi ])d\xi }$ . 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

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 ${\displaystyle \sigma }$ , ${\displaystyle \sigma }$  can be characterized by a vector field ${\displaystyle \delta (r;\sigma )}$ . ${\displaystyle \delta (r;\sigma )=0}$  for all positions ${\displaystyle r\notin \sigma }$ . For all positions ${\displaystyle r\in \sigma }$ , ${\displaystyle \delta (r;\sigma )}$  is infinite in the direction of the outwards normal direction to ${\displaystyle \sigma }$  in a manner similar to the Dirac delta function. The integral property that must be satisfied by ${\displaystyle \delta (r;\sigma )}$  is that for any oriented path ${\displaystyle C}$ , if ${\displaystyle C}$  passes through ${\displaystyle \sigma }$  in the preferred direction a net total of ${\displaystyle N}$  times, then

${\displaystyle \int _{r\in C}\delta (r;\sigma )\bullet dr=N}$

(${\displaystyle C}$  passing through ${\displaystyle \sigma }$  in the reverse direction decreases ${\displaystyle N}$  by 1.)

Given any vector field ${\displaystyle F(r)}$ , ${\displaystyle \int _{r\in \sigma }F(r)\bullet dA=\iiint _{r\in \mathbb {R} ^{3}}(F(r)\bullet \delta (r;\sigma ))d\tau }$

It is easy to verify that if ${\displaystyle \sigma }$  is a closed surface, then ${\displaystyle \delta (r;\sigma )}$  is irrotational.

Given any sequence of surfaces ${\displaystyle \sigma _{1},\sigma _{2},\dots ,\sigma _{k}}$ , these surfaces can be added in a linear fashion to get a "multi-surface" denoted by the vector field ${\displaystyle \delta (r;\sigma _{1})+\delta (r;\sigma _{2})+\dots +\delta (r;\sigma _{k})}$ . The multi-surface is denoted by: ${\displaystyle \sigma _{1}+\sigma _{2}+\dots +\sigma _{k}}$ .

Most importantly, given any irrotational vector field ${\displaystyle F}$  that decreases faster than ${\displaystyle o(1/|r|^{2})}$  as ${\displaystyle |r|\rightarrow +\infty }$ , then there exists a family ${\displaystyle \sigma [\xi ]}$  of closed surfaces where ${\displaystyle \xi \in D_{\sigma }}$  is an arbitrary continuous indexing parameter such that ${\displaystyle F(r)=\iint _{\xi \in D_{\sigma }}\delta (r;\sigma [\xi ])d\xi }$ . In simpler terms, any irrotational vector field can be expressed as a linear combination of closed surfaces.

Given an oriented surface ${\displaystyle \sigma }$  with a counter-clockwise oriented boundary ${\displaystyle C}$ , it is then the case that ${\displaystyle \nabla \times \delta (r;\sigma )=\delta (r;C)}$ . Given any vector field ${\displaystyle F}$  that denotes a multi-surface, then ${\displaystyle \nabla \times F}$  is a vector field that denotes the counter-clockwise oriented boundary of the multi-surface denoted by ${\displaystyle F}$ . 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

Let ${\displaystyle C_{1}}$  and ${\displaystyle C_{2}}$  be two oriented closed loops, and let ${\displaystyle \sigma _{1}}$  and ${\displaystyle \sigma _{2}}$  be oriented surfaces whose counter-clockwise boundaries are respectively ${\displaystyle C_{1}}$  and ${\displaystyle C_{2}}$ .

Given a current of ${\displaystyle I}$  flowing around ${\displaystyle C_{1}}$ , let ${\displaystyle B_{1}}$  be the magnetic field induced via Ampere's law. Note that ${\displaystyle B_{1}(r)\propto I}$ . The magnetic flux through surface ${\displaystyle \sigma _{2}}$  is

${\displaystyle \Phi _{B,2}=\iint _{r\in \sigma _{2}}B_{1}(r)\bullet dA}$  where ${\displaystyle dA}$  is the vector representation of an infinitesimal surface element of ${\displaystyle \sigma _{2}}$ .

Note that also, ${\displaystyle \Phi _{B,2}\propto I}$ . This constant of proportionality, ${\displaystyle M_{1,2}={\frac {\Phi _{B,2}}{I}}}$ , is the mutual electromagnetic induction from ${\displaystyle C_{1}}$  to ${\displaystyle C_{2}}$ .

The mutual electromagnetic induction from ${\displaystyle C_{1}}$  to ${\displaystyle C_{2}}$  will be denoted with ${\displaystyle M(C_{1},C_{2})}$

### Self Inductance

When ${\displaystyle C_{2}=C_{1}}$ , the inductance ${\displaystyle L(C_{1})=M(C_{1},C_{1})}$  is referred to as the "self inductance".

### Linearity of Mutual Inductance

Given loops ${\displaystyle C_{1}}$ , ${\displaystyle C_{2}}$ , and ${\displaystyle C_{3}}$ , it is relatively simple to demonstrate that ${\displaystyle M(C_{1}+C_{3},C_{2})=M(C_{1},C_{2})+M(C_{3},C_{2})}$  and ${\displaystyle M(C_{1},C_{2}+C_{3})=M(C_{1},C_{2})+M(C_{1},C_{3})}$ .

Let ${\displaystyle B_{1}(r)}$ , ${\displaystyle B_{2}(r)}$ , and ${\displaystyle B_{3}(r)}$  be the magnetic fields generated when a current of ${\displaystyle I}$  flows through ${\displaystyle C_{1}}$ , ${\displaystyle C_{2}}$ , or ${\displaystyle C_{3}}$  respectively.

The magnetic field generated by ${\displaystyle C_{1}}$  and ${\displaystyle C_{3}}$  together is ${\displaystyle B_{1}(r)+B_{3}(r)}$  due to the linearity of Maxwell's equations. This leads to ${\displaystyle M(C_{1}+C_{3},C_{2})=M(C_{1},C_{2})+M(C_{3},C_{2})}$ .

The flux through ${\displaystyle C_{2}+C_{3}}$  is the sum of the flux through ${\displaystyle C_{2}}$  and ${\displaystyle C_{3}}$  separately. This leads to ${\displaystyle M(C_{1},C_{2}+C_{3})=M(C_{1},C_{2})+M(C_{1},C_{3})}$ .

### Symmetry of Mutual Inductance

It is the case that given loops ${\displaystyle C_{1}}$  and ${\displaystyle C_{2}}$ , that ${\displaystyle M(C_{1},C_{2})=M(C_{2},C_{1})}$ . 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 ${\displaystyle C_{1}}$  and ${\displaystyle C_{2}}$  will be interpreted as multi-surfaces whose boundaries are respectively ${\displaystyle C_{1}}$  and ${\displaystyle C_{2}}$ .

Let there exist a current of ${\displaystyle I}$  in loop ${\displaystyle C_{1}}$ , and let ${\displaystyle B_{1}(r)}$  denote the resultant magnetic field. Ampere's law requires that ${\displaystyle \nabla \times B_{1}(r)=\mu _{0}I\delta (r;C_{1})\implies \nabla \times {\frac {1}{\mu _{0}}}{\frac {B_{1}(r)}{I}}=\delta (r;C_{1})}$ , and therefore ${\displaystyle {\frac {1}{\mu _{0}}}{\frac {B_{1}(r)}{I}}}$  is a multi-surface whose boundary is ${\displaystyle C_{1}}$ . Since ${\displaystyle B_{1}(r)\propto I}$ , let ${\displaystyle b_{1}(r)={\frac {B_{1}(r)}{I}}}$ .

Given a divergence free vector field ${\displaystyle F}$ , the flux of ${\displaystyle F}$  through ${\displaystyle \sigma _{1}}$  is:

${\displaystyle \iint _{r\in \sigma _{1}}F(r)\bullet dA=\iiint _{r\in \mathbb {R} ^{3}}F(r)\bullet \delta (r;\sigma _{1})d\tau =\iiint _{r\in \mathbb {R} ^{3}}F(r)\bullet {\frac {b_{1}(r)}{\mu _{0}}}d\tau }$

The final equality holds due to the fact that ${\displaystyle F}$  is divergence free and that ${\displaystyle \delta (r;\sigma _{1})}$  and ${\displaystyle {\frac {b_{1}(r)}{\mu _{0}}}}$  are multi-surfaces with a common boundary of ${\displaystyle C_{1}}$ .

${\displaystyle B_{1}(r)}$  is divergence free. The flux of ${\displaystyle B_{1}(r)}$  through ${\displaystyle \sigma _{2}}$  is:

${\displaystyle \Phi _{B,2}=\iiint _{r\in \mathbb {R} ^{3}}I{\frac {b_{1}(r)\bullet b_{2}(r)}{\mu _{0}}}d\tau }$

Therefore: ${\displaystyle M(C_{1},C_{2})=\iiint _{r\in \mathbb {R} ^{3}}{\frac {b_{1}(r)\bullet b_{2}(r)}{\mu _{0}}}d\tau }$  from which the symmetry ${\displaystyle M(C_{1},C_{2})=M(C_{2},C_{1})}$  is now apparent.

## Calculating the Mutual Inductance

### Approach #1 (use the vector potential)

Gauss' law of magnetism requires that ${\displaystyle \nabla \bullet B=0}$ . This makes possible a "vector potential" for ${\displaystyle B}$ : a vector field ${\displaystyle A}$  which satisfies ${\displaystyle \nabla \times A=B}$ . The condition ${\displaystyle \nabla \bullet A=0}$  can also be enforced.

Using the vector identity:

For any vector field ${\displaystyle F}$ : ${\displaystyle \nabla \times (\nabla \times F)=\nabla (\nabla \bullet F)-\nabla ^{2}F}$

Ampere's law becomes:

${\displaystyle \nabla \times B=\mu _{0}j\iff \nabla \times (\nabla \times A)=\mu _{0}j\iff \nabla (\nabla \bullet A)-\nabla ^{2}A=\mu _{0}j\iff \nabla ^{2}A=-\mu _{0}j}$

${\displaystyle \nabla ^{2}A=-\mu _{0}j}$  is an instance of Poisson's equation which has the solution: ${\displaystyle A(r)={\frac {\mu _{0}}{4\pi }}\iiint _{r'\in \mathbb {R} ^{3}}{\frac {j(r')}{|r-r'|}}d\tau '}$

It can be checked that for this solution, since ${\displaystyle \nabla \bullet j=0}$ , that ${\displaystyle \nabla \bullet A=0}$ .

The vector potential generated by a current of ${\displaystyle I}$  flowing through closed loop ${\displaystyle C_{1}}$  is: ${\displaystyle A_{1}(r)={\frac {\mu _{0}}{4\pi }}\iiint _{r_{1}\in \mathbb {R} ^{3}}{\frac {I\delta (r_{1};C_{1})}{|r-r_{1}|}}d\tau _{1}={\frac {\mu _{0}}{4\pi }}\int _{r_{1}\in C_{1}}{\frac {I}{|r-r_{1}|}}dr_{1}}$

The magnetic field generated by a current of ${\displaystyle I}$  flowing through closed loop ${\displaystyle C_{1}}$  is: ${\displaystyle B_{1}=\nabla \times A_{1}}$ . The flux through surface ${\displaystyle \sigma _{2}}$  (which is counter-clockwise bounded by ${\displaystyle C_{2}}$ ), is

${\displaystyle \Phi _{B,2}=\iint _{r_{2}\in \sigma _{2}}B_{1}(r_{2})\bullet dA_{2}=\iint _{r_{2}\in \sigma _{2}}(\nabla \times A_{1})(r_{2})\bullet dA_{2}=\int _{r_{2}\in C_{2}}A_{1}(r_{2})\bullet dr_{2}}$  via Stoke's theorem.

${\displaystyle \Phi _{B,2}={\frac {\mu _{0}}{4\pi }}\int _{r_{2}\in C_{2}}\int _{r_{1}\in C_{1}}{\frac {I}{|r_{2}-r_{1}|}}(dr_{1}\bullet dr_{2})}$  so the mutual inductance is: ${\displaystyle M(C_{1},C_{2})={\frac {\Phi _{B,2}}{I}}={\frac {\mu _{0}}{4\pi }}\int _{r_{2}\in C_{2}}\int _{r_{1}\in C_{1}}{\frac {dr_{1}\bullet dr_{2}}{|r_{2}-r_{1}|}}}$

This equation is known as "Neumann formula" [1].

It can also be seen from this expression that the mutual inductance is symmetric: ${\displaystyle M(C_{1},C_{2})=M(C_{2},C_{1})}$ .

### Approach #2 (use linearity and loop dipoles)

Given any closed loop ${\displaystyle C}$ , let ${\displaystyle \sigma }$  be an oriented surface that has ${\displaystyle C}$  as its counterclockwise boundary. For each infinitesimal area vector element ${\displaystyle dA}$  of ${\displaystyle \sigma }$ , let the infinitesimal ${\displaystyle \partial (dA)}$  be an infinitesimal closed loop that is the counterclockwise boundary of ${\displaystyle dA}$ . It is then the case that ${\displaystyle C=\int _{r\in \sigma }\partial (dA)}$ .

The linearity of the mutual inductance gives:

${\displaystyle M(C_{1},C_{2})=M\left(\iint _{r_{1}\in \sigma _{1}}\partial (dA_{1}),\iint _{r_{2}\in \sigma _{2}}\partial (dA_{2})\right)=\iint _{r_{1}\in \sigma _{1}}\iint _{r_{2}\in \sigma _{2}}M(\partial (dA_{1}),\partial (dA_{2}))}$

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 ${\displaystyle A}$ , and a current ${\displaystyle I}$  that flows around the boundary of ${\displaystyle A}$  in a counterclockwise manner, then the magnetic dipole (vector) formed is ${\displaystyle P=IA}$ . If the area shrinks, then the current increases proportionally if the magnetic dipole is to remain constant.

Given a magnetic dipole ${\displaystyle P}$  with an infinitesimal area at position ${\displaystyle 0}$ , the magnetic field produced by ${\displaystyle P}$  is:

${\displaystyle B(r)={\frac {\mu _{0}}{4\pi }}\left({\frac {3(P\bullet r)r}{|r|^{5}}}-{\frac {P}{|r|^{3}}}\right)}$

Let ${\displaystyle a_{1}}$  and ${\displaystyle a_{2}}$  be area vectors of the interiors of two infinitesimal loops, with the second loop displaced from the first by ${\displaystyle r}$ . Let a current ${\displaystyle I}$  flow around the boundary of ${\displaystyle a_{1}}$  in a counter clockwise manner forming the dipole ${\displaystyle Ia_{1}}$ . The flux of the magnetic field generated by ${\displaystyle Ia_{1}}$  through ${\displaystyle a_{2}}$  is:

${\displaystyle \Phi _{B,2}={\frac {\mu _{0}I}{4\pi }}\left({\frac {3(a_{1}\bullet r)(a_{2}\bullet r)}{|r|^{5}}}-{\frac {a_{1}\bullet a_{2}}{|r|^{3}}}\right)}$

Therefore if ${\displaystyle c_{1}}$  and ${\displaystyle c_{2}}$  are the counter clockwise boundaries of ${\displaystyle a_{1}}$  and ${\displaystyle a_{2}}$ :

${\displaystyle M(c_{1},c_{2})={\frac {\mu _{0}}{4\pi }}\left({\frac {3(a_{1}\bullet r)(a_{2}\bullet r)}{|r|^{5}}}-{\frac {a_{1}\bullet a_{2}}{|r|^{3}}}\right)}$

Returning to computing the mutual inductance between ${\displaystyle C_{1}}$  and ${\displaystyle C_{2}}$  gives:

${\displaystyle M(C_{1},C_{2})={\frac {\mu _{0}}{4\pi }}\iint _{r_{1}\in \sigma _{1}}\iint _{r_{2}\in \sigma _{2}}\left({\frac {3(dA_{1}\bullet (r_{2}-r_{1}))(dA_{2}\bullet (r_{2}-r_{1}))}{|r_{2}-r_{1}|^{5}}}-{\frac {dA_{1}\bullet dA_{2}}{|r_{2}-r_{1}|^{3}}}\right)}$

This formula is centered around surface integrals as opposed to loop integrals.

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