Saved Bookmarks
| 1. |
Draw a ray diagram showing the formation of the image by a point object on the principal axis of a spherical convex surface separating two media of refractive indices n_(1) and n_(2), when apoint source is kept in rarer medium of refractive index n_(1). Derive the relation between object and image distance in terms of refractive index of the medium and radium of curvature of the surface. Hence obtain the expression for lens-maker's formula in the case of thin convex lens. |
Answer» Solution : The incident rays coming from the OBJECT .O. kept in the rarer medium of REFRACTIVE index `n_(1)`, incident on the REFRACTING SURFACE NM produce the real image at I. From the DIAGRAM, `/_ i = /_ NOM + /_ NCM` `= ( NM )/( OM )+ ( NM )/( MC )` `/_r = /_ NCM - /_ NIM ` `= ( NM )/( MC ) - ( NM ) /( MI )` From Snell.s law, `:. ( n_(2))/( n_(1)) = ( sin i)/( sin r ) ~ (i) /( r )`( for small angles, sin` theta ~ theta )` `:. n_(2) r = n_(1) i ` or `n_(2) ((NM)/( MC) - ( NM )/( MI)) = n_(1) ((NM)/( OM) + ( NM)/( MC))` or `n_(2) ((1)/( + R ) - ( 1)/( +v)) = n_(1) ((1)/( - u ) + ( 1)/( R ))` or `(n_(2) - n_(1))/( R ) = ( n_(2))/( v ) - ( n_(1))/( u )` Lens maker.s formula `:`  The first refracting surface ABC forms the image `I_(1)` of the object O. The image `I_(1)` acts as a virtual object for the second refracting surface ADCwhich forms the real image I as shown in the diagram. `:`. For refraction at ABC `(n_(2))/( v_(1)) - ( n_(1))/(u)= ( n_(2) - n_(1))/( R_(1))`....(i) For refraction at ADC `(n_(1))/( v) - ( n _(2))/(v_(1)) = ( n_(1) - n_(2))/( R_(2))`....(ii) Adding equations (i) and (ii) , we get `(n_(1))/( v) - ( n_(1))/( u ) = ( n _(2) - n_(1)) ((1)/( R_(1)) - ( 1)/( R_(2)))` `(1)/( v) - ( 1)/( u ) = ( ( n_(2))/(n_(1)) - 1) ((1)/( R_(1)) - ( 1)/( R_(2)))` `(1)/( f) = ( mu_(21) - 1) ((1)/( R_(1))- ( 1)/( R_(2)))` |
|