Prove 4f is the Min. Distance Between Conjugate Points for Thin Lens

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fluidistic
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Homework Statement


Show that the minimum distance between 2 conjugate points (real object and image) for a positive thin lens is 4f.

Homework Equations


[tex]\frac{1}{f}=\frac{1}{S_o}+\frac{1}{S_i}[/tex].

The Attempt at a Solution


I assumed the lens to be biconvex (though I know that I can't. There are so many types of positive lens...).
So I get that [tex]\frac{1}{f}=(n_1-n_0)\left ( \frac{2}{R} \right )[/tex].So I must show that [tex]S_0+S_1 \geq 4f[/tex].
Using these 2 formulae, I reach that the inequation holds if and only if [tex]S_0+S_i \leq 2R[/tex] where R is the curvature radius of the thin lens. I'm stuck here. Are there any other equation I should use? Or am I in the right direction?
 
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Hi fluidistic! :smile:

You're making this far too complicated!

Just write Si - So as a function of So, and differentiate. :wink:
 
tiny-tim said:
Hi fluidistic! :smile:

You're making this far too complicated!

Just write Si - So as a function of So, and differentiate. :wink:

Wow, this worked. Awsome! So I only needed the formula [tex]\frac{1}{f}=\frac{1}{S_o}+\frac{1}{S_i}[/tex] and your nice idea!:biggrin:

Edit: Wait! Why did you choose the expression So-Si rather than minimizing So+Si?
 
fluidistic said:
Edit: Wait! Why did you choose the expression So-Si rather than minimizing So+Si?

oops! :rolleyes:
 
tiny-tim said:
oops! :rolleyes:

But this worked! ahahahaha, I made an error but I reached the result. Wow, amazing. I'll retry. Ahahahah.
 
Have you solved the problem? I'm getting stuck, I just don't reach anything. [tex]S_0+S_i=\frac{S_0S_i}{f}[/tex]. I have to minimize this function.
 
No, So + Si = So + 1/(1/f - 1/So) = So + fSo/(So - f) …

carry on from there. :smile:

(and I'm off to bed :zzz:)​
 
tiny-tim said:
No, So + Si = So + 1/(1/f - 1/So) = So + fSo/(So - f) …

carry on from there. :smile:

(and I'm off to bed :zzz:)​


Worked! Thanks a lot.
 
It is still unclear to me where to go from here. What should I be differentiating with respects to?

Regards,
Adam
 
Hi Adam! :smile:
Titans86 said:
What should I be differentiating with respects to?

hmm … there's only one variable in the formula …
tiny-tim said:
No, So + Si = So + 1/(1/f - 1/So) = So + fSo/(So - f)

… so i suppose you'd better differentiate wrt that! :wink: