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Transcript
PHY 042: Electricity and Magnetism Maxwell’s equations Prof. Hugo Beauchemin 1 Limitation of Ampere’s law With Faraday’s law, we extended the electrostatics E-field to non- steady current, providing a set of equations for E-fields valid in any classical situations and describing all possible experiments However, there is still an opened question: Is Ampere’s law still valid when currents are non-steady? No, as is, Ampere’s law is limited to statics situations Static case! Maxwell spotted this limitation and proposed an extension: Use continuity equation to add a term guaranteeing that the divergence of the curl of B is always null for any current 2 The displacement current The extra term added by Maxwell to yield divergenceless curl for the B-field in non-static situation changed Ampere’s law: A changing electric field induces a magnetic field! Maxwell’s term was named displacement current: It restores charge conservation in Ampere’s law for non-static case All magnetostatics results obtained are preserved Brought a complete symmetry between the E-field and the B-field Hard to observe, so not in disagreement with previous experiments, but can be tested once it is expected/predicted Purely theoretical addition, making new predictions for experiments in non-static situation! 3 The typical example Consider a circuit with a charging up capacitor Assume plates are large compared to their separation Use infinite capacitor plate approximation Assume weak steady current No induced EMF and uniform static charge distribution on plates These small assumptions need to be carefully controlled in a real experiment to allow for a measurement of the wanted effects By Biot-Savart, the B-field due to current in iii) and iv) is: B must be continuous at the boundary of region iii) and ii) No plate with surface current between regions ii) and iii) iii) No wire can yield a B-field in ii) Ampere’s law on JD must be added to satisfy boundary conditions ii) i) iv) 4 Maxwell equations We have a set of differential equations completely determining: How charges produce an E-field and a B-field How a varying E-field or B-field generates the other field How the fields affect the motion of charges Differential form Integral form i) ii) iii) iv) v) 5 Few comments (I) 1. The continuity equation, expressing the charge conservation, is a consequence of iv) and can be obtained from the divergence of iv) 2. To solve actual problems (make predictions for experiments), we need to know the boundary conditions Boundary conditions are exactly the same as those found in electroand magneto-statics: 3. If we are dealing with materials, it is convenient to write Maxwell’s equations in terms of free charges and currents Note that if the E-field varies in time, so will the polarization, thus generating a flow of bound charges 6 Few comments (II) Maxwell’s equations are written in terms of B, H, D, E and must thus be supplemented with constitutive equations and Typically only given in terms of E and H Control potentials and currents 4. The large symmetry between E and B is understood in special relativity as the fact that E and B are related by Lorentz transform. This complete the unification of E and B For a boost along x-axis 7 Few comments (III) The two curl equations introduce a coupling between the two fields and this is fundamental to the understanding of electromagnetic wave propagating in vacuum (J=r=0) Taking the curl of Faraday’s and Ampere-Maxwell’s laws: This is the exact form of the propagation of progressive wave functions The speed of this wave is v = (m0e0)-1 = 2.998 × 108 m/s = c These waves are light!!!! 8