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Maxwells 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
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Contents1. Introduction2. Gauss’s law for electric fields3. Gauss’s law for magnetic fields4. Faraday’s law5. The Ampere-Maxwell law December 07 Bruna Larissa Lima Crisóstomo 2
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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.December 07 Bruna Larissa Lima Crisóstomo 3
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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.”December 07 Bruna Larissa Lima Crisóstomo 4
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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 contributesReminder that this is a The electric An increment ofsurface integral (not a The electric permittivity field in N/C surface area in m²volume or line integral) of the space December 07 Bruna Larissa Lima Crisóstomo 5
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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 divergenceDecember 07 Bruna Larissa Lima Crisóstomo 6
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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.”December 07 Bruna Larissa Lima Crisóstomo 7
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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 ofReminder that this is a induction in surface area in m²surface integral (not a Teslasvolume or line integral) December 07 Bruna Larissa Lima Crisóstomo 8
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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 divergenceDecember 07 Bruna Larissa Lima Crisóstomo 9
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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.”December 07 Bruna Larissa Lima Crisóstomo 10
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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 changeTells you to sum up The electric of the magneticthe contributions field in N/C induction with timefrom each portionof the closed path Reminder that this is a line The rate of changeC integral (not a surface or a with time volume integral) December 07 Bruna Larissa Lima Crisóstomo 11
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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 curlDecember 07 Bruna Larissa Lima Crisóstomo 12
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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.”December 07 Bruna Larissa Lima Crisóstomo 13
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The Ampere-Maxwell law Integral formReminder that the Dot product tells you to findmagnetic field is a the part of H parallel to dlvector (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 fluxTells you to sum up the contributions through a surface Reminder that onlyfrom each portion of the closed path C bounded by C the enclosed currentin direction given by ruth-hand rule contributes December 07 Bruna Larissa Lima Crisóstomo 14
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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 fieldReminder that the with timedell operator is avector 𝜕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 December 07 Bruna Larissa Lima Crisóstomo 15
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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 LucianoDecember 07 Bruna Larissa Lima Crisóstomo 16
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Reference FLEISCH, DANIEL A. A Student’s Guide to Maxwell’s Equations. First published. United States of America by Cambrige University Press, 2008.December 07 Bruna Larissa Lima Crisóstomo 17
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