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# Maxwell's equations

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

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### Maxwell's equations

1. 1. 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
2. 2. 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
3. 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.December 07 Bruna Larissa Lima Crisóstomo 3
4. 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.”December 07 Bruna Larissa Lima Crisóstomo 4
5. 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 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
6. 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 divergenceDecember 07 Bruna Larissa Lima Crisóstomo 6
7. 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.”December 07 Bruna Larissa Lima Crisóstomo 7
8. 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 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
9. 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 divergenceDecember 07 Bruna Larissa Lima Crisóstomo 9
10. 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.”December 07 Bruna Larissa Lima Crisóstomo 10
11. 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 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
12. 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 curlDecember 07 Bruna Larissa Lima Crisóstomo 12
13. 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.”December 07 Bruna Larissa Lima Crisóstomo 13
14. 14. 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
15. 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 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
16. 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 LucianoDecember 07 Bruna Larissa Lima Crisóstomo 16
17. 17. 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