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Consider a plane wavefront of electromagnetic fields travelling with a speed c in the right (say+Z) direction, it is given that vecE and vecB are transverse to each other and uniform throughoutthe left of the wavefront and zero on the right of the wavefront. [This is contrived, but not incorrect, configuration chosen for simplicity. In the usual monochromatic plane wave, vecE and vecB very sinusoidally in space and time]. (a) Use Faraday's law to show that E=cB. (b) Use Ampere's law (with displacement current included) to show that c=1//sqrt(mu_0in_0) |
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Answer» Solution :Let `vecE` be in the x-direction and `vecB` in the y- direction. (a) Consider the rectangular loop in the XZ plane with ONE side of LENGTH l prallel to `vecE`. At the instant under consideration, the rectangle is PARTIALLY on left and partially on the RIGHT of the wavefront. Rate of change of magnetic flux =Blc The line integral of `vecE=oint vec E.vec(dl)=El`. From Faraday's laws of electromagnetic induction `:. El=Blcor E=Bc....(i)` (b) Consider a similar rectangle in the YZ plane. Rate of change of electric flux=Elc. The line integral of `vecB` is Bl. from Ampere's law or `Bl=mu_0in_0Elc` or `B=mu_0in_0Ec` or `B=mu_0in_0c(Bc) ` [from EQ(i)] or `c^2=1/(mu_0in_0) or c=1/(sqrt(mu_0in_0))` |
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