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 (bar (E). bar (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
`int (bar (E). bar (dl)) = E xx 2pi r`. As `phi_(B)= (B_(0)t) pi r^(2)`
` therefore (d phi_(B))/(dt)= B_(0) pir^(2)`
` therefore E xx 2pi r = -B_(0) pir^(2)`
`E= (-B_(0)r)/(2) ( E prop r)`

(2) `r gt R`. Consider a ring of radius `r (gt R)` centred at O. Again following the same procedure
`int (bar (E). bar (dl)) = E xx 2pi r`
`phi_(B)= (B_(0)t)pi R^(2)` ( as the field is only present in region `0 lt r lt R`)
`(d phi_(B))/(dt)= B_(0) pi R^(2)`
`implies E2pir= - B_(0)pi R^(2)`
`E=( -B_(0)R^(2))/(2r) (implies E prop (1)/(r))`