Uma apresentação feita para o grupo PET-Elétrica UFCG no dia 07 de dezembro de 2012.

1. 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

2. Contents
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
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3. Introduction
 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.
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4. Gauss’s law for electric fields
 Integral form:
𝑞 𝑒𝑛𝑐
𝐸 h 𝑛 𝑑𝑎 =
𝑆 𝜀0
“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:
𝜌
𝛻h 𝐸 =
𝜀0
“The electric field produced by electric charge diverges from positive charges
and converges from negative charges.”
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5. Gauss’s law for electric fields
Integral form
Reminder that the
Dot product tells you to find the part of E
eletric field is a
parallel to n (perpendicular to the surface)
vector
The unit vector normal The amount of
Reminder that this
to the surface change in coulombs
integral is over a
closed surface 𝑞 𝑒𝑛𝑐
𝐸 h𝑛 𝑑𝑎 = Reminder that only
𝑆 𝜀0 the enclosed charge
contributes
Reminder that this is a
The electric An increment of
surface integral (not a The electric permittivity
field in N/C surface area in m²
volume or line integral) of the space
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6. Gauss’s law for electric fields
Differential form
Reminder that the electric
Reminder that field is a vector The electric charge
del is a vector density in coulombs
operator per cubic meter
𝜌
𝛻h𝐸 =
The differential
operator called
𝜀0 The electric
permittivity of free
“del” or “nabla” space
The electric
field in N/C
The dot product turns
the del operator into
the divergence
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7. Gauss’s law for magnetic fields
 Integral form:
𝐵 h 𝑛 𝑑𝑎 = 0
𝑆
“The total magnetic flux passing through any closed surface is zero.”
 Differential form:
𝛻h 𝐻 = 0
“The divergence of the magnetic field at any point is zero.”
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8. Gauss’s law for magnetic fields
Integral form
Reminder that the Dot product tells you to find
magnetic field is a the part of B parallel to n
vector (perpendicular to the surface)
The unit vector normal to the surface
Reminder that this
integral is over a
closed surface
𝐵 h𝑛 𝑑𝑎 = 0
𝑆
The magnetic An increment of
Reminder that this is a induction in surface area in m²
surface integral (not a Teslas
volume or line integral)
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9. Gauss’s law for magnetic fields
Differential form
Reminder that the magnetic
Reminder that field is a vector
del is a vector
operator
The differential
𝛻h𝐻 = 0
operator called
“del” or “nabla” The magnetic
field in A/m
The dot product turns
the del operator into
the divergence
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10. Faraday’s law
 Integral form:
𝑑
𝐸h 𝑑 𝑙 = − 𝐵h 𝑛 𝑑𝑎
𝐶 𝑑𝑡 𝑠
“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:
𝜕𝐵
𝛻×𝐸 = −
𝜕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.”
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11. Faraday’s law
Integral form
Dot product tells you to find The magnetic flux
Reminder that the the part of E parallel to dl through any surface
eletric field is a (along parth C) bounded by C
An incremental segment of path C
vector
𝑑
𝐸 h𝑑 𝑙 = − 𝐵h𝑛 𝑑𝑎
𝐶 𝑑𝑡 𝑠 The rate of change
Tells you to sum up The electric of the magnetic
the contributions field in N/C induction with time
from each portion
of the closed path Reminder that this is a line The rate of change
C integral (not a surface or a with time
volume integral)
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12. Faraday’s law
Differential form
Reminder that the electric
Reminder that field is a vector
del is a vector
operator
The rate of change
𝜕𝐵 of the magnetic
𝛻×𝐸 = −
induction with time
The differential
operator called 𝜕𝑡
“del” or “nabla” The electric
field in V/m
The cross-product
turns the del
operator into the curl
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13. The Ampere-Maxwell law
 Integral form:
𝑑
𝐻h 𝑑 𝑙 = 𝐼 𝑒𝑛𝑐 + 𝜀0 𝐸h 𝑛 𝑑𝑎
𝐶 𝑑𝑡 𝑠
“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:
𝜕𝐸
𝛻×𝐻 = 𝐽 + 𝜀0
𝜕𝑡
“The circulating magnetic field is produced by any electric current and by an
electric field that changes with time.”
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14. The Ampere-Maxwell law
Integral form
Reminder that the Dot product tells you to find
magnetic field is a the part of H parallel to dl
vector (along path C) The rate of change
An incremental The electric current with time
segment of path in amperes
C
𝑑
𝐻h𝑑 𝑙 = 𝐼 𝑒𝑛𝑐 + 𝜀0 𝐸h𝑛 𝑑𝑎
𝐶 𝑑𝑡 𝑠
The electric
The magnetic
permittivity of
field in A/m
free space
The electric flux
Tells you to sum up the contributions through a surface
Reminder that only
from each portion of the closed path C bounded by C
the enclosed current
in direction given by ruth-hand rule
contributes
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15. The Ampere-Maxwell law
Differential form
Reminder that the Reminder that the The electric
magnetic field is a current density is a permittivity of The rate of change
vector vector free space of the electric field
Reminder that the with time
dell operator is a
vector
𝜕𝐸
𝛻×𝐻 = 𝐽 + 𝜀0
𝜕𝑡
The differential
operator called
“del” or “nabla”
The magnetic
field in A/m The electric current density
in amperes per square
The cross-product turns meter
the del operator into the
curl
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16. Maxwell’s Equations
brunallcrisostomo@gmail.com
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 16

17. Reference
 FLEISCH, DANIEL A. A Student’s Guide to Maxwell’s Equations. First
published. United States of America by Cambrige University Press,
2008.
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