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

AI Thread Summary
The discussion revolves around proving that the minimum distance between two conjugate points for a positive thin lens is 4f. Participants explore the lens formula and attempt to derive the relationship between object distance (S_o) and image distance (S_i). One user suggests simplifying the problem by expressing S_i - S_o as a function of S_o and differentiating it, which leads to a successful resolution of the problem. The conversation highlights the importance of correctly applying the lens formula and the differentiation technique to find the minimum distance. Ultimately, the users confirm that they reached the desired result through collaboration and problem-solving.
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


\frac{1}{f}=\frac{1}{S_o}+\frac{1}{S_i}.

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 \frac{1}{f}=(n_1-n_0)\left ( \frac{2}{R} \right ).So I must show that S_0+S_1 \geq 4f.
Using these 2 formulae, I reach that the inequation holds if and only if S_0+S_i \leq 2R 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 \frac{1}{f}=\frac{1}{S_o}+\frac{1}{S_i} 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. S_0+S_i=\frac{S_0S_i}{f}. 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
 
  • #10
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:
 

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