The focal length of a mirror is given by #1/f=1/u+1/v# where u and v represent object and image distances respectively,The maximum relative error in f is ?

1 Answer
Feb 23, 2018

Given expression is

#1/f=1/u+1/v#
#=>1/f=(v+u)/(uv)#
#=>f=(uv)(v+u)^-1#

Taking #log# of both sides we get

#logf=logu+logv+log(v+u)^-1#
#=>logf=logu+logv-log(v+u)#

Differentiating with respect to each variable.

#(Deltaf)/f=(Deltau)/u+-(Deltav)/v+-(Delta(u+v))/(v+u)#
Here #Deltaf# represents error in #f# and so on. It is assumed that errors in each are independent, random, and sufficiently small

#=>(Deltaf)/f=(Deltau)/u+-(Deltav)/v+-(Deltau+Deltav)/(v+u)#
#=>(Deltaf)/f=(Deltau)/u+-(Deltav)/v+-(Deltau)/(v+u)+-(Deltav)/(v+u)#

We calculate maximum average error as below

#Deltaf=barf+-sqrt(((Deltau)/u)^2+((Deltav)/v)^2+((Deltau)/(v+u))^2+((Deltav)/(v+u))^2)#