A uniform magnetic field B exists in a cylindrical region, shown dotted in. The magnetic field increases at a constant rate `(dB)/(dt)` Consider a circle of radius r coaxial with the cylindrical region. (a) find the magnitude of the electric field E at a point on the circumference of the circle. (b) Consider a point P on the side of the square circumscribing the circle. Show that the component of the induced electric field at P along ba is the same as the magnitude found in part(a).

Explanation

(a) work done per unit per unit test change is `
`phi.E.dl=e implies E phi .dl=(d phi)/(dt)implies E. 2 pi r =(dB)/(dt)xxA`
`implies E 2 pi r = pi r^2(dB)/(dt)`
`implies E= (pi r^2)/(2 pi r) (dB)/(dt) =(r/2)(dB)/(dt)`
(b)when the square is considered, `
phi.e.dt=e`
`implies Exx2rxx4=(dB)/(dt)(2r)^2`
`implies E = (dB)/(dt) ((4r^2)/(8r)`
` impliesE=(r/2) (dB)/(dt). `
`(c) the electric field at the point has the same value as (a)`.