A uniform magnetic field exists in a circular region of radius R centrad 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

As we know, induced electric field is given by
`intE.Dl=(-dphi_(B))/(dt)`
(i) For, `r lt R`. Consider a ring of radius `r(ltR)` centered at O.
The induced field at the periphery of the ring is circular such that,
`ointE.dl=Exx2pir`…..(i) (`because`dl = circumference of the ring `=2pir`
As, magnetic flux, `phi_(B)=B.S=BScostheta`
Here, `theta=0^(@)," so "phi_(B)=(B_(0)t)pir^(2)" "{beacuseB=B_(0)t]`

`therefore" "(dphi_(B))/(dt)=B_(0)pir^(2)`....(ii)
Equating Eqs. (i) and (ii), we get
`therefore" "Exx2pir=-B_(0)pir^(2)`
`E=(-B_(2)r)/(2)rArrEprop r`
(the sign comes, when clockwise electric field lines are developed)
(ii) For, `r gt R`. Consider a ring of radius `r(gtR)` centred at O.
Again following same procedure,
`ointE.dl=Exx2pir`...(iii)
Similarly, flux, `phi_(B)=(B_(0)t)piR^(2)` (as field is present only in region `0 lt r lt R).`......(iv)

`rArr" "(dphi_(B))/(dt)=B_(0)piR^(2)`
Equating Eqs. (iii) and (iv), we get
`rArr" "E2pir=-B_(0)piR^(2)`
`rArr" "E=(-B_(0)R^(2))/(2r)rArrE prop(1)/(r)`
Hence, plot between `|E|` and r will be