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)