A) \[\frac{R+l}{R-l}\]
B) \[{{\left( \frac{R+l}{R-l} \right)}^{2}}\]
C) \[\frac{2\,(R-l)}{R+l}\]
D) \[2{{\left( \frac{R-l}{R+1} \right)}^{2}}\]
Correct Answer: B
Solution :
For a spherical refracting surface, |
\[\frac{\mu }{v}-\frac{1}{\left( -u \right)}=\frac{\mu -1}{R}\]\[\Rightarrow \frac{\mu }{v}+\frac{1}{\left( u \right)}=\frac{\mu -1}{R}\] |
Let \[u+v=x\] |
\[\Rightarrow \,\,\,\frac{\mu }{x-u}+\frac{1}{u}=\frac{\mu -1}{R}\]\[\Rightarrow \,\,\,x=u+\frac{\mu Ru}{\left( \mu -1 \right)u-R}\] |
\[\therefore \frac{du}{dx}=\frac{\mu Ru}{{{\left( \left( \mu -1 \right)u-R \right)}^{2}}}\] |
L is minimum, when \[\frac{du}{dx=0}\] |
\[\therefore u=\frac{R}{\sqrt{\left( \mu \right)}-1}\] |
From Eq. (i), we have |
\[v=\frac{\sqrt{\mu }\,\,R}{\sqrt{\mu }-1}\] |
As \[l=u+v,l=\frac{R}{\sqrt{\mu }-1}+\frac{\sqrt{\mu }.R}{\sqrt{\mu }-1}\] |
\[\mu ={{\left( \frac{R+l}{R-l} \right)}^{2}}\] |
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