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Show how to generalize Ampere's circuital law to include the term due to displacement current? |
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Answer» Solution :According to Ampere's circuital law, `underset(s)( oint )vec(B)cdot vec(dl)= mu_(0)` As ther current flows across the area bounded by loop `S_(1)`,so `underset(s_(2))(oint )vec(B)cdot vec(dl)= mu_(0)`. But the area bounded by `S_(2)` lies in the REGION between the plates capacitor where no current flows across it. `underset(s_(2))(oint) vec(B)cdot vec(dl) =0.` Consider that loops enclosing S1 & S2 are infinitesimally close to each other. then `underset(s_(1))(oint )vec(B)cdot vec(dl)= underset(s_(2))(oint )vec(B)cdot vec(dl)` This equation is INCONSISTENT with equation (2) & (3). to remove this maxwell said that a CHANGING electric field (during charging ) between the capacitor plates MUST induce a magnetic field which in turn must be associated with current `I_(d)`. `I_(d) = epsilon_(0)((d phi_(E))/(dt))[ (d phi_(E))/(dt) " change in electric flux"]` The total current must be `I = I_("conduction ") + I_("displacement")` `I_(c )= epsilon_(0) (d phi_(E))/(dt)` Hence the generalised from of Ampere's circuital law is `oint vec(B)cdot vec(dl)= mu_(0) [ I_(C ) + epsilon_(0) (d phi_(E))/(dt) ]` |
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