A short object of length L is placed along the principal axis of a concave mirror away from focus. The object distance is u. If the mirror has a focal length f, what will be the length of the image ? You may take `L lt lt |v- f|.`

Explanation

Distance of closect end of an object (of length L ) from the pole of concave mirror,
`u_1 = u + 1/2`
`rArr` Similarly distance of farthest end of above object from the pole of concave mirror,
`mu_2 = u +L/2`
`therefore u_1 - u_2 = - L`
`therefore | mu_1 - mu_2| = L`
If image distance for above two ends are repectively `v_1` and `v_2` then length of image will be `L. = |v_1 - v_2|`
`rArr` From mirror formula
`1/f = 1/u + 1/v` we have `1/v = 1/f - 1/u`
`therefore 1/v = (u-f)/(fu)`
`rArr` From above equation
`therefore = (fu)/(u-f)`
`v_ = (f(u-L/2))/(u-L/2-f) and v_2 = (f(u+L/2))/(u+L/2-f)`
`therefore ` Length of image
`L. = | v_1 - v_2|`
`= (fu-(fL)/(2))/(u-f-L/2) - (fu+(fL)/(2))/(u-f + L/2)`
` =1/((u-f)^(2) - L^2/4)xx [ fu^2-f^2u + (fuL)/(2)-(fuL)/(2) + (f^2L)/(2)-(fL^2)/(4)-fu^2 + f^2u + (fuL)/(2) - (fuL)/(2) + (f^2L)/(2) + (fL^2)/(4)]`
`= (f^2L)/((u-f^2)-L^2/4)`
`rArr` Hence we have `(L^2)/(4) lt lt (u - f)^2` and neglecting `(L^2)/(4)` from the denominator,
`therefore L. = |v_1 - v_2| = (f^2)/((u-f)^2)L`