Prozorov_19 - PHYSICS 222 Introduction to Classical Physics...

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Unformatted text preview: PHYSICS 222 Introduction to Classical Physics II Prof. Ruslan Prozorov Iowa State University Fall 2011 LECTURE 19 The Maxwell Equations James Clerk Maxwell (1831–1879) Born 13 June 1831 Edinburgh, Scotland Died 5 November 1879 (aged 48) Cambridge, England Citizenship United Kingdom Nationality Scottish In the millennium poll—a survey of the 100 most prominent physicists — Maxwell was voted the third greatest physicist of all time, behind only Newton and Einstein. Maxwell, J. Clerk, “A Dynamical Theory of the Electromagnetic Field”, Philosophical Transactions of the Royal Society of London, volume 155, pages 459-512, 1865, doi: 10.1098/rstl.1865.0008 PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 2 the equations • Maxwell Equations, in their modern form of four partial differential equations, first appeared in fully developed form in his textbook “A Treatise on Electricity and Magnetism” in 1873. • Maxwell expressed electromagnetism in the algebra of quaternions and made the electromagnetic potential the centerpiece of his theory. • In 1881 Oliver Heaviside replaced Maxwell’s electromagnetic potential field by ”force fields” as the centerpiece of electromagnetic theory. Heaviside reduced the complexity of Maxwell’s theory down to four differential equations, known now collectively as Maxwell's Laws or Maxwell's equations. PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 3 Oliver Heaviside (18 May 1850 – 3 February 1925) was a selftaught English electrical engineer, mathematician, and physicist who adapted complex numbers to the study of electrical circuits, invented mathematical techniques to the solution of differential equations (later found to be equivalent to Laplace transforms), reformulated Maxwell's field equations in terms of electric and magnetic forces and energy flux, and independently co-formulated vector analysis. Although at odds with the scientific establishment for most of his life, Heaviside changed the face of mathematics and science for years to come. PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 4 A Treatise on Electricity and Magnetism PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 5 problem with the Ampere’s law B dl I 0 enclosed when capacitor is charging/discharging dQ t dV Q t CV t , I C dt dt there is NO wire inside? V where does the current come from? Q t CV t 0 A 0 AE 0 E There is electric field (flux) d dQ t dE magnetic field is created by moving charge ID 0 dt dt and by time-varying electric field! Maxwell postulated the “displacement current” dE B dl 0 I 0 0 dt PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 6 displacement current Although current is flowing through the capacitor, no actual charge is transported through the vacuum between its plates. Nonetheless, a magnetic field exists between the plates as though a current were present there as well. The explanation is that a displacement current ID flows in the vacuum, and this current produces the magnetic field in the region between the plates according to Ampère's law. PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 7 elements of differential math ˆ ˆ ˆ x y z x y z - del operator (nabla symbol) 2 2 2 2 2 2 2 x y z divB B - divergence curlB B - Laplacian - curl PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 8 The Maxwell Equations PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 9 in the absence of sources there is an impressive symmetry of the equations B E 0 E t E B 0 B 0 0 t PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 10 E produces B, B produces E Example: At some point P in space, the measured electric field as a function of time is: E P t This E field MUST be accompanied by a B field. But if the induced B field also changes with time, it induces an E field! This can be self-sustained! “Perturbation” (E or B field) propagates in space wave Very special wave, can propagate in vacuum: no medium! PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 11 the problem is in material Constitutive relations In order to apply 'Maxwell's equations', it is necessary to specify the relations between displacement field D and E, and the magnetic H-field and B-field. These equations specify the response of bound charge and current to the applied fields (response of the material) and are called constitutive relations. The definition of the auxiliary fields: D r , t 0E r , t P r , t H r, t 1 0 B r, t M r, t PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 12 The Maxwell Equations PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 13 example of the use of Maxwell eqs V1 a) battery V1 R2 V2 R1 R1 B R2 V2 in both cases we will have currents through resistors b) solenoid with changing field PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 14 in vacuum (no materials) D r , t 0E r , t H r, t 1 0 B r, t PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 15 isotropic linear materials PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 16 example of the use of Maxwell eqs electrons in a superconductor (no scattering!) dv d j ne 2 eE, j nev E m dt m dt d ne 2 ne 2 B j E dt m m t ne 2 ne 2 B 0 j B0 j t m m B E t B 0 j B 0 Maxwell equations 0 j B B B B B 0 ne 2 m B 0 B B 0, 2 London equation PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University m 0 ne 2 London penetration depth 7 October 2011 17 solution of the London equation d 2B B B z , 0, 0 2 2 B 0 dz d2 d2 d2 2 2 2 2 dx dy dz B B 0 2 z B B0 exp so, as expected lim B z 0 z PHYS222 - Lecture 19 - Prof. Ruslan Prozorov - Iowa State University 7 October 2011 18 ...
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This note was uploaded on 11/14/2011 for the course PHYS 5863005 taught by Professor Meyer during the Fall '09 term at Iowa State.

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