Maxwell's Equations of Electrodynamics an Explanation [2012]
Maxwell's equations
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Transcript of 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
December 07 Bruna Larissa Lima Crisóstomo 5
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
December 07 Bruna Larissa Lima Crisóstomo 6
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 electric field in N/C
The dot product turns the del operator into the divergence
The differential operator called “del” or “nabla”
December 07 Bruna Larissa Lima Crisóstomo 7
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.”
December 07 Bruna Larissa Lima Crisóstomo 8
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
December 07 Bruna Larissa Lima Crisóstomo 9
Gauss’s law for magnetic fields Differential form
𝛻h𝐻 = 0
Reminder that del is a vector operator
Reminder that the magnetic field is a vector
The magnetic field in A/m
The dot product turns the del operator into the divergence
The differential operator called “del” or “nabla”
December 07 Bruna Larissa Lima Crisóstomo 10
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.”
December 07 Bruna Larissa Lima Crisóstomo 11
Faraday’s law Integral form
𝐸 h𝑑𝑙 𝐶
= −𝑑
𝑑𝑡 𝐵h𝑛 𝑑𝑎𝑠
Reminder that the eletric field is a vector
Dot product tells you to find
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
The electric field in N/C
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
December 07 Bruna Larissa Lima Crisóstomo 12
Faraday’s law Differential form
𝛻×𝐸 = −𝜕𝐵
𝜕𝑡
Reminder that del is a vector operator
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 differential operator called “del” or “nabla”
The rate of change of the magnetic induction with time
December 07 Bruna Larissa Lima Crisóstomo 13
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.”
December 07 Bruna Larissa Lima Crisóstomo 14
The Ampere-Maxwell law Integral form
𝐻h𝑑𝑙 = 𝐼𝑒𝑛𝑐 + 𝜀0𝑑
𝑑𝑡 𝐸h𝑛 𝑑𝑎𝑠𝐶
Reminder that the magnetic field is a vector
Dot product tells you to find
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
The electric permittivity of free space
Reminder that only the enclosed current contributes
The electric flux through a surface bounded by C
December 07 Bruna Larissa Lima Crisóstomo 15
The Ampere-Maxwell law Differential form
𝛻×𝐻 = 𝐽 + 𝜀0𝜕𝐸
𝜕𝑡
The cross-product turns the del operator into the curl
The differential operator called “del” or “nabla”
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
The electric permittivity of free space
Reminder that the magnetic field is a vector
Reminder that the current density is a vector
Reminder that the dell operator is a vector
December 07 Bruna Larissa Lima Crisóstomo 16
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 Bruna Larissa Lima Crisóstomo 17
Reference
FLEISCH, DANIEL A. A Student’s Guide to Maxwell’s Equations. First
published. United States of America by Cambrige University Press,
2008.