The minimum distance between an object and it's real image

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SUMMARY

The minimum distance between an object and its real image in geometric optics, specifically using the thin lens equation, is established as 4f, where f represents the focal length of the lens. To prove this mathematically, one must derive an equation for the total distance (d1 + d2) in terms of d1, differentiate it with respect to d1, and solve for the value of d1 that results in a derivative of zero. This process confirms that the minimum distance is indeed 4f.

PREREQUISITES
  • Understanding of geometric optics principles
  • Familiarity with the thin lens equation
  • Basic calculus, including differentiation
  • Knowledge of focal length in optics
NEXT STEPS
  • Study the derivation of the thin lens equation
  • Learn about the properties of real images in optics
  • Explore calculus applications in physics, particularly differentiation
  • Investigate advanced topics in geometric optics, such as lens combinations
USEFUL FOR

Students in physics, optical engineers, and anyone interested in the mathematical foundations of lens behavior in geometric optics.

eep
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Maybe it's just late, but I'm having an extremely difficult time proving that the minimum distance between an object and it's real image (geometric optics, thin lense equation) is 4f. I can see that it is true, however I'm unsure how to go about proving it mathematically.
 
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write an equation for the total distance d1+d2 in term of just d1 then differentiate it wrt d1 find value of d1 for which dif = zero. then find d1+d2
 

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