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## James Clerk Maxwell

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**Symbols**E = Electric field ρ = charge density i = electric current B = Magnetic field εo = permittivity J = current density D = Electric displacement μo = permeability c = speed of light H = Magnetic field strength M = Magnetization P = Polarization**q**Gauss’s Law N No magnetic monopoles S B ∂E/∂t Ampere’s Circuital Law i E Faraday’s Law of Induction ∂B/∂t**Maxwell took all the semi-quantitative conclusions of**Oersted, Ampere, Gauss and Faraday and cast them all into a brilliant overall theoretical framework. The framework is summarised in Maxwell’s Four Equations**Maxwell’s Equations**. E = 0 ∆ .B = 0 ∆ E = - (∂B/∂t) x ∆ B = μoεo(∂E/∂t) x ∆**Feynman on Maxwell'sContributions**"Perhaps the most dramatic moment in the development of physics during the 19th century occurred to J. C. Maxwell one day in the 1860's, when he combined the laws of electricity and magnetism with the laws of the behavior of light.**From a long view of the history of mankind — seen from,**say, ten thousand years from now, there can be little doubt that the most significant event of the 19th century will be judged as Maxwell's discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade.**http://en.wikipedia.org/wiki/File:KL_Kernspeicher_Makro_1.jpg**http://en.wikipedia.org/wiki/File:KL_Kernspeicher_Makro_1.jpg**1nm diametre 1mm diametre**10-9 m 10-2 m ca 10,000,000 smaller**These equations are a bit complicated and we are not going**to deal with them in this very general course. However we can discuss arguably the most important and at the time most amazing consequence of these equations.**N**B = μoεo ∂E/∂t x ∆**N**S E = - ∂B/∂t ∆ x**As equations are combined – for instance when one has two**equations in twounknowns one can juggle the equations and obtain two new equations each involving only one of the unknowns and so solve them.**εo = dielectric constant or pemittivity**μo = permeability**1**μoεo= c2**Maxwell’s Equations - it’s hard to get a shot of all**four at once - it’s hard to get my arm in the right position for any camera to get Ampere’s circuital law.” Weldon Harry Kroto 2004**As a result, the properties of light were partly unravelled**-- that old and subtle stuff that is so important and mysterious that it was felt necessary to arrange a special creation for it when writing Genesis. Maxwell could say, when he was finished with his discovery, 'Let there be electricity and magnetism, and there is light!' " Richard Feynman in The Feynman Lectures on Physics, vol. 1, 28-1.**Maxwell took all the semi-quantitative conclusions of**Oersted, Ampere, Gauss and Faraday and cast them all into a brilliant overall theoretical framework. The framework is summarised in Maxwell’s Four Equations**2-D Array of Ni6 clusters**with Prashant Jain Naresh Dalal and Tony Cheetham**Calculus**Differentiation**Calculus**Differentiation dy/dx = y**Calculus**Differentiation dy/dx = y y = ex**Calculus**Differentiation dy/dx = y y = ex eix = cosx + isinx**dsinx/dx = cosx**and dcosx/dx = - sinx**dsinx/dx = cosx**and dcosx/dx = - sinx thus d2sinx/dx2 = -sinx**These equations are a bit complicated and we are not going**to deal with them in this very general course. However we can discuss arguably the most important and at the time most amazing consequence of these equations.**. E = 0**.B = 0 ∆ ∆ B = μoεo(∂E/∂t) x E = - (∂B/∂t) ∆ ∆ x**x(**xE) = xE) x(- ∂B/∂t) = -(∂/∂t)( ∆ ∆ ∆ ∆ x(- ∂B/∂t) = ∆ x( xE) = xB) -(∂/∂t)( ∆ ∆ ∆**x(**xE) = xE) x(- ∂B/∂t) = -(∂/∂t)( ∆ ∆ ∆ ∆ x( xE) = xB) -(∂/∂t)( ∆ ∆ ∆ -(∂/∂t) = μo εo (∂E/∂t) = = - μo εo (∂E2/∂t2)**Lesson 19 - Ampere's Law As Modified by Maxwell**I. Capacitor Problem C In the figure below, we attempt to apply Ampere's law for a wire leading to a capacitor using the curve, C. IcIc Although it is easy to write Ampere's Law, we find that we have a paradox since the value for the integral depends on the current that penetrates any surface bounded by the curve. For a circular membrane, the current passing through the surface is Ic. For a surface that wraps around the capacitor, we have no current penetrating the surface. In 1865, James Clerk Maxwell, one of the great physicists of all times, solved this paradox and developed electromagnetic theory (one of the major branches of physics). II. Ampere's Law As Modified By Maxwell The current flowing through the wire supplies the charge on the plates of the capacitor that produces the electric field across the capacitor plates. From our previous work with parallel plate capacitors, we have From the definition of current, we have**Maxwell's contribution was to imagine that the changing**electric flux between the capacitor was equivalent to a physical current as far as the creation of a magnetic field. He called this "fictitious" current the displacement current. To remove the paradox, Maxwell equated the displacement current to the current in the wire and modified the right hand side of Ampere's Law to include the sum of the real current and the displacement current. In our work, there is no difference for our parallel plate capacitor between the partial derivative of the electric flux with respect to time and the full time derivative of the electric flux. However, Maxwell using more powerful mathematical techniques solved the problem in general thereby showing that it is the partial derivative of the electric flux with respect to time. Thus, we have written our final result so that it will be correct for all problems. IMPORTANT: This incredible result states that there is a second way to create a circulating magnetic field: A time varying electric flux!! We will return later in the course to this wonderful result and its importance in communications.**N**B = μoεo ∂E/∂t x ∆