InterviewSolution
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InterviewSolution
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The four Maxwell's equations and the Lorentz force law (which together constitution the fundations of all the classical electromagnetism) are listed below: (i) oint vecB.vec(ds)=q//(in_0) (ii) oint vecB.vec(ds)=0 (iii) oint vecE.vec(dl)=-d/(dt) int_svecB.vec(ds) (iv) oint vecB.vec(dl)=mu_0I+mu_0d/(dt)int_s vecb.vec(ds) Lorentz force law: vecF=q(vecE+vecvxxvecB) Answer the following question regarding these equation: (a) Give the name (s) associated with some of the four equation above. (b) Which equations above contain source vecE and vecB and which do not? what do the equations reduce to in a source-free region? (c) Write down Maxwell's equations for steady (i.e. time independent) electric and magnetic fields. (d) If magnetic monopoles existed, which of the equations would be modified? Suggest how they might be modified? (e) Which of the four equations shown that magnetic field lines cannot start from a point nor end at a point? (f) Which of the four equations show that electrostatic field lines cannot form closed loops? (g) The equations listed above refer to integrals of vecE and vecBover loops/surfaces Can we write down equations for vecE and vecB for each point in space? (h) Are the equations listed above true for different types of media: dielectrics, conductors, plasmas etc.? (i) Are the equation true fora arbitrarily high and low values of vecE,vecB,q,I? |
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Answer» Solution :(a) (i) Guss's law in electrostatics. (ii) Guss's law in magnetostatics. (iii) Faraday's law of electromagnetic induction (iv) Ampere's Circutal law with Maxwell's modification. (b) Equations (i) and (iv) contain the source Q, I: equations (ii) and (iii)do not. To obtain equations in source free region, simply put `q=0` and `I=0`. Then `oint vecE.vec(ds)=0`[from eqn(i)] and `oint vecE.vec(dl)=mu_0in_0d/(dt)int_svecE.vec(ds)` [From eq(iv)] (c) Maxwell's equations will be TIME independent if the derivative of the physical quantity involved with time is zero. Putting this concept on the right hand side of EQUATION (iii) and (iv), we have Maxwell's equations as: `oint vecE.vec(ds)=Q/(in_0)oint vecE.vec(ds)=0` `oint vecE.vec(dl)=0oint vecE.vec(dl)=mu_0I` (d) Equation (ii) and (iii) would be modified . Equation (ii) is based on the fact that monopoles do not exist. If the monopoles exist, the right hand side would contain a term say `q_m` representing magntic dipole strength, analogous to Gauss's law, in electrostatics, we would have `oint vecE.vec(ds)=q_mxxcostant.` Further equation (iii) would also be modified. An additional term `I_m` representing the current due to flow of magnetic charge would have to be included on the right hand side of equation (iii) analogous to the electric charge current of equation (iv), we would have `oint vecE.vec(dl)=I_mxxconstant-d/(dt)int_svecB.vec(ds)` All this is of COURSE, based on the expection of symmetry of from of the equaitons for E and B. Nature may never show up monopoles or else even if monopoles exist, the actual modification of Maxwell's equations might be very different. (e) Equation (ii) only (f) Equation (i) shows that the elctrostatic field lines cannot from closed loops, as electrostatic field field lines cannot pass through conductors. (g) Yes, we can. Maxwell's equations can be cast as different equations valid at every POINT in space and at every instant. (h) Maxwell's equations are true in all media. But in macroscopic media, it is usually convenient to write down equations for average of `vecE and vecB` over regions which are small macroscopically but large enough to contain a very large number of atoms etc. The resulting macroscopic Maxwell's equtions are of great practical use. (i) Maxwell's equations are the basic laws of classical electromagnetism. They are true in all media and for any value of E,B,q,I ect. whithin the domain of validity of classical electromagnetism. The precise domain of validity is hard to specify and need not concern us here. |
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