The ratio of the intensity at the cnetre of a bright fringe to the intensity at a point one quarter of the fringe width from the centre is

Options

A. `2`
B. `1//2`
C. `4`
D. `16`

Explanation

Two waves of a single source having an amplitude `A` interfere. The resulting amplitude
`A_(r)^(2)=A_(1)^(2)+A_(2)^(2)+2A_(1)A_(2)Cosdelta`
where `A_(1)=A_(2)=A` and `delta=` phase difference between the waves
`rArr I_(r)=I_(1)+I_(2)+2sqrt(I_(1)I_(2))Cosdelta`
When the maxima occurs at the center, `delta=0`
`rArr I_(r_(1))=4I`.....(`1`)
Since the phase difference between, two successive frings is `2pi`, the phase difference between two points separated by a distance equal to one quarter of the distance between the two, successive frings is equal to
`delta=(2pi)((1)/(4))=(pi)/(2)radian`
`rArrI_(r_(2))=4Icos^(2)((pi//2)/(2))=2I`....(`2`)
Using Eqs. (`1`) and (`2`),
`(I_(r_(1)))/(I_(r_(2)))=(4I)/(2I)=2`