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 << |v-f|
As the mean distance of object from mirror is u ∴ u1=u−2L and u2=(u+2L) Let the image of the two ends of object form at distance v1 and v2(v1>v2). So lenght of image on principle axis is L′(v1−v2) v1+u1=f1 or v1+f1=u1 ⇒ v1=ufu−f⇒v=u−fuf So L′=v1−v2==(u−2L)−f(u−2L)f−(u+2L)−f(u+2L)f⇒L′=f(u−f−2L)u−2L−(u−f+2L)u+21 =f⎣⎢⎢⎢⎢⎢⎡(u−f−2L)(u−f+2L)(u−2L)(u−f+2L)−(u+2L)(u−f−2L)⎦⎥⎥⎥⎥⎥⎤ =(u−f)2−4L2f[u2−uf+2uL−2uL+2fL−4L2−(u2−uf+2uL−2uL+2fL−4L2)] ∴L<<(u−f) So neglecting the terms 4L2 L2=f⎣⎢⎢⎢⎡(u−f)22fL+2fL⎦⎥⎥⎥⎤ L′=(u−f)2ffL⇒L′=(u−f)2Lf2 it is the length of image f
