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Maxwell's equations

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2Contents1. 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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3Introduction In Maxwell’s equations there are:

the eletrostatic field produced by electric charge;

the induced field produced by changing magnetic field.

Donot confuse the magnetic field (𝐻) withdensity 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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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= 𝜀 0“The electric field produced by electric charge diverges from positive charges and converges from negative charges.”

Gauss’s law for electric fields

∫𝑆

❑

𝐸 .�̂� .𝑑 �⃗�=𝑞𝑒𝑛𝑐  𝜀0

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5Gauss’s law for electric fieldsIntegral form

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6Gauss’s law for electric fieldsDifferential form

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7Gauss’s law for magnetic fields

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8Gauss’s law for magnetic fieldsIntegral form

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9Gauss’s law for magnetic fieldsDifferential 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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10Faraday’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“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 thedirection so as to oppose the change in flux.”

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11Faraday’s lawIntegral form

Dot product tells you to find the part of E parallel to d l (along parth C)

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12Faraday’s lawDifferential form

𝛻×𝐸 = - 𝜕𝐵𝜕𝑡

Reminder that del is a vector operator

Reminder that the electric field is a vector

The electric field in V/mThe 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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13The Ampere-Maxwell law

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14The Ampere-Maxwell lawIntegral form

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15The Ampere-Maxwell lawDifferential form

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16Reference FLEISCH, DANIEL A. A Student’s Guide to Maxwell’s Equations. First

published. United States of America by Cambrige University Press,

2008.