In Young's double slit experiment two rectangular slits are used.What will happen if instead of two rectangular slits two concentric circular slits(of small radii) are used,the distance between their circumferences remaining as it is between the two rectangular slits.How will the pattern observed on the screen change or will we not observe any interference at all?
The pattern will have the same symmetry as the slits: it will consist of something like a bull's eye pattern. The exact arrangement of light and dark circular fringes will depend on the widths of the two annular (circular) slits and their radii.
If you would like to figure it out for yourself, you can think of it this way: Unroll the annular slits into a pair of linear slits. Add another pair of linear slits, parallel to the first pair, and offset by twice the radius of each slit.
Note that the two pairs of slits is equivalent to a stretched-out version of a cross-section of the double annular slit. The interference pattern from the two pairs of linear slits would be the coherent superposition of the two individual interference patterns that each pair would produce: a linear fringe pattern of the same spacing as one pair produces, modulated by a very short-period linear fringe pattern corrresponding to the spacing between the two pairs.
Now "roll up" the linear fringe pattern to form a bull's eye pattern. This is an approximation of the interference pattern you would get from your double-ring interferometer. An exact calculation can be done if you're handy with calculus.