InterviewSolution
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InterviewSolution
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A roundcurrent-carryinglooplies in the plane boundary between magnetic and vacumm. The permeabilityof th emagnetic is equal to mu Find themagnetic induction Bat an arbitary point on the axis of the loop if in the absence of th e magnetic the magneticinduction at thesame pointbecomesequal to B_(0) . Genralize theobtainedresult to all poinsof the field. |
Answer» Solution :The medium `I` is vacumm and contains a circular current carryingcoil withcurrent `I`. The medium `II` is a megenticwith permeabitliy`mu`. The boundaryisthe plane`z = 0` and the coil is the plane`z = 1`. To findthe magnitudeinduction we notethat theeffectof the magenticmediumcan bewrittenas DUE to an image coil in `II` as faras the medim `I` isconcerned. Onthe other hand, theinduction is `II` as far asthe medium`I` isconcerned, Onthe other hand, the induction is `II` can bewrittenas due t the coil in `I`, carryinga differentcurrent. It issufficientto considerthe far awayfieldsand ensure that the BOUNDARY conditionsare SATISFIED there. Now for actualcoil in medium`I`. `B_(R) = - (2p_(m) cos theta')/(r^(3)) . ((mu_(0))/(4pi)). B_(0) = (p_(m) sin theta')/(r^(3)) ((mu_(0))/(4pi))` so,`B_(2) = (mu_(0) P m)/(4pi) (2 cos^(2) theta' - sin^(2) theta')` and`B_(x) = (mu_(0) p_(m))/(4pi) (-3 sin theta' cos theta')` where `p_(m) = I (PI a^(2)), a` = radiusof the coil. Similarlydue to the image coil, `B_(x) = (mu_(0) p')/(4pi) (2 cos^(2) theta' - sin^(2) theta), B_(x) = (mu_(0) p')/(4pi) (3 sin theta' cos theta'), p'_(m) = I' (pi a^(2))` As fas as the medium`II` is concered, we write similarly `B_(z) = (mu_(0) p''_(m))/(4pi) (2 cos^(2) - sin^(2) theta'), B_(x) = (mu_(0) p''_(m))/(4pi) (-3 sin theta' cos theta'). p''_(m)= I''(pi a^(2))` The boundaryconditionsare, `p_(m) + p'_(m) = p"_(m)` (from `B_(1n) - B_(2n)`) `p_(m) + p'_(m) = - (1)/(mu) p"_(m)` (from`H_(1t) = H_(2t)`) Thus,`I" = (2 mu)/(mu+ 1) I, I' = (mu - 1)/(mu + 1) I` Inthe limit, whenthe coil is on the boundary, the magnetic filedenvery where can beobtained by takingthe currenttobe `(2mu)/(mu + 1) I`. Thus, `vec(B) = (2mu)/(mu + 1) vec(B_(0))` |
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