Maxwell's equations

Maxwell's equations Universidade Federal de Campina Grande

Centro de Engenharia Elétrica e Informática

Departamento de Engenharia Elétrica

Programa de Educação Tutorial – PET -Elétrica

Student Bruna Larissa Lima Crisóstomo

Tutor Benedito Antonio Luciano

December 07

Contents

Bruna Larissa Lima Crisóstomo 2

1. Introduction

2. Gauss’s law for electric fields

3. Gauss’s law for magnetic fields

4. Faraday’s law

5. The Ampere-Maxwell law

December 07

Introduction

Bruna Larissa Lima Crisóstomo 3

In Maxwell’s equations there are:

the eletrostatic field produced by electric charge;

the induced field produced by changing magnetic field.

Do not confuse the magnetic field (𝐻) with density

magnetic (𝐵), because 𝐵 = 𝜇𝐻. 𝐵 : the induction magnetic or density magnetic in Tesla; 𝜇: the permeability of space ;

𝐻 : the magnetic field in A/m.

December 07 Bruna Larissa Lima Crisóstomo 4

Integral form:

𝐸 h𝑛 𝑑𝑎𝑆

= 𝑞𝑒𝑛𝑐𝜀0

“Electric charge produces an electric field, and the flux of that field passing through any closed surface is proportional to the total charge contained within that surface.”

Differential form:

𝛻h𝐸 =𝜌

𝜀0

“The electric field produced by electric charge diverges from positive charges and converges from negative charges.”

Gauss’s law for electric fields


Gauss’s law for electric fields Integral form

𝐸 h𝑛 𝑑𝑎𝑆

= 𝑞𝑒𝑛𝑐𝜀0

Reminder that this integral is over a closed surface

Reminder that the eletric field is a vector

Dot product tells you to find the part of E parallel to n (perpendicular to the surface)

The unit vector normal to the surface

The amount of change in coulombs

Reminder that only the enclosed charge contributes

The electric permittivity of the space

An increment of surface area in m²

Reminder that this is a surface integral (not a volume or line integral)

The electric field in N/C

𝛻h𝐸 =𝜌

𝜀0


Gauss’s law for electric fields Differential form

Reminder that del is a vector operator

Reminder that the electric field is a vector The electric charge

density in coulombs per cubic meter

The electric permittivity of free space


The dot product turns the del operator into the divergence

The differential operator called “del” or “nabla”


Gauss’s law for magnetic fields

Integral form:

𝐵 h𝑛 𝑑𝑎𝑆

= 0

“The total magnetic flux passing through any closed surface is zero.”

Differential form: 𝛻h𝐻 = 0

“The divergence of the magnetic field at any point is zero.”


Gauss’s law for magnetic fields Integral form

𝐵 h𝑛 𝑑𝑎𝑆

= 0

Reminder that the magnetic field is a vector

Dot product tells you to find

the part of B parallel to n (perpendicular to the surface)

The unit vector normal to the surface

An increment of surface area in m²

The magnetic induction in Teslas

Reminder that this is a surface integral (not a volume or line integral)

Reminder that this integral is over a closed surface


Gauss’s law for magnetic fields Differential form

𝛻h𝐻 = 0



The magnetic field in A/m

The dot product turns the del operator into the divergence



Faraday’s law

Integral form:

𝐸 h𝑑 𝑙 𝐶

= −𝑑

𝑑𝑡 𝐵h𝑛 𝑑𝑎𝑠

“Changing magnetic flux through a surface induces a voltage in any boundary path of that surface, and changing the magnetic flux induces a circulating electric field.“

Differential form:

𝛻×𝐸 = −𝜕𝐵

𝜕𝑡

“A circulating electric field is produced by a magnetic induction that changes with time.“

Lenz’s law: “Currents induced by changing magnetic flux always flow in the direction so as to oppose the change in flux.”


Faraday’s law Integral form

𝐸 h𝑑𝑙 𝐶

= −𝑑

𝑑𝑡 𝐵h𝑛 𝑑𝑎𝑠

Reminder that the eletric field is a vector


the part of E parallel to dl (along parth C)

An incremental segment of path C

The magnetic flux through any surface bounded by C

The rate of change of the magnetic induction with time


Reminder that this is a line integral (not a surface or a volume integral)

Tells you to sum up the contributions from each portion of the closed path C

The rate of change with time


Faraday’s law Differential form

𝛻×𝐸 = −𝜕𝐵

𝜕𝑡


Reminder that the electric field is a vector

The electric field in V/m

The cross-product turns the del operator into the curl


The rate of change of the magnetic induction with time


The Ampere-Maxwell law

Integral form:

𝐻h𝑑 𝑙 = 𝐼𝑒𝑛𝑐 + 𝜀0𝑑

𝑑𝑡 𝐸h𝑛 𝑑𝑎𝑠𝐶

“The electric current or a changing electric flux through a surface produces a circulating magnetic field around any path that bounds that surface.”

Differential form:

𝛻×𝐻 = 𝐽 + 𝜀0𝜕𝐸

𝜕𝑡

“The circulating magnetic field is produced by any electric current and by an electric field that changes with time.”


The Ampere-Maxwell law Integral form

𝐻h𝑑𝑙 = 𝐼𝑒𝑛𝑐 + 𝜀0𝑑

𝑑𝑡 𝐸h𝑛 𝑑𝑎𝑠𝐶



the part of H parallel to dl (along path C)

An incremental segment of path C

The electric current in amperes

The rate of change with time

Tells you to sum up the contributions from each portion of the closed path C in direction given by ruth-hand rule

The magnetic field in A/m


Reminder that only the enclosed current contributes

The electric flux through a surface bounded by C


The Ampere-Maxwell law Differential form

𝛻×𝐻 = 𝐽 + 𝜀0𝜕𝐸

𝜕𝑡

The cross-product turns the del operator into the curl


The magnetic field in A/m The electric current density

in amperes per square meter

The rate of change of the electric field with time



Reminder that the current density is a vector

Reminder that the dell operator is a vector


Maxwell’s Equations

Universidade Federal de Campina Grande

Centro de Engenharia Elétrica e Informática

Departamento de Engenharia Elétrica

Programa de Educação Tutorial – PET -Elétrica

Student Bruna Larissa Lima Crisóstomo

Tutor Benedito Antonio Luciano

[email protected]

mailto:[email protected]


Reference

FLEISCH, DANIEL A. A Student’s Guide to Maxwell’s Equations. First

published. United States of America by Cambrige University Press,

2008.

Maxwell's equations

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