A uniform magnetic field exists in a circular region of radius R centred at O. The field is perpendicular to the plane of paper and is strength varies with time as `B=B_(0)t`. Find the induced electric field at a distance r from the centre for (i)`r lt R`, (ii) `r gt R`. Also, plot a graph between `|E|` and r for both the cases.

Explanation

Induced electric field is given by
`int vecE. Vec(dl)=(-d phi_(B))/(dt)`
(1) r lt R. Consider a ring of radius r (lt R) centered at O. The induced field at the periphery of the ring is circular such that
`ointvecE.vec(dl)=Exx2pir". As "phi_(B)=(B_(0)t)pir^(2)`
`therefore (dphi_(B))/(dt)=B_(0)pir^(2)`
`therefore Exx2pir=-B_(0)r^(2)`
`E=(-B_(0)r)/(2) (rArr E prop r)`
(the sign comes when clockwise electric field lines are developed)

(2) r gt R. Consider r ring of radius r ( gt R) centred at O. Again following the same procedure
`ointvecE. vec (dl)=Exx2pir`
`phi_(B)=(B_(0)t)piR^(2)" (as field is present only in rigion 0 lt r lt R)"`
`(dphi_(B))/(dt)=B_(0)piR^(2)`
`rArr E2pir=-B_(0)piR^(2)`
`E=(-B_(0)R^(2))/(2r)" "(rArr E prop (1)/(r))`