Maxwell's equations describe the fundamental interactions between electricity and magnetism. They include: 1) Gauss's law for electric fields, which relates the electric flux through a closed surface to the electric charge enclosed. 2) Gauss's law for magnetic fields, which states that the magnetic flux through a closed surface is always zero, since there are no magnetic monopoles. 3) Faraday's law, which describes how a changing magnetic field induces an electric field. It relates the circulating electric field to the rate of change of the magnetic field. 4) The Ampere-Maxwell law, which describes how electric currents and changing electric fields generate magnetic fields. It relates the magnetic field to the electric current

1. Maxwell's equations
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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.  Integral form:
“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
𝛻.E=
𝜀0
“The electric field produced by electric charge diverges from positive charges
and converges from negative charges.”
Gauss’s law for electric fields
𝑆
𝐸. 𝑛 . 𝑑 𝑎 =
‫ݍ‬݁݊ܿ
𝜀0
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5. Gauss’s law for electric fields
Integral form
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6. Gauss’s law for electric fields
Differential form
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7. Gauss’s law for magnetic fields 7

8. Gauss’s law for magnetic fields
Integral form
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9. Gauss’s law for magnetic fields
Differential form
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”
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10. Faraday’s law
 Integral form:
“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
“Acirculating 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 part of E parallel to d l
(along parth C)
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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
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13. The Ampere-Maxwell law 13

14. The Ampere-Maxwell law
Integral form
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15. The Ampere-Maxwell law
Differential form
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16. 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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