Isaacphysics Deriving the Lensmaker's Equation

Question Link: https://isaacphysics.org/questions/deriving_lensmaker

Hi, I am just stuck on part D as I am not sure how to derive the lensmaker's equation using the information given. From the previous parts and the information given in part D, I have found a few expressions which are relevant:
$R_1 sin(\mu_1) = h$
$R_2 sin(\mu_2) = h$
$u tan(\theta_1) = h$
$v tan(\theta_2) = h$
$\mu_2 = (\theta_1+\theta_2)/(n-1) - \mu_1$

I have also found these expressions form the previous parts (which I dont think are relevant but will still add):
$\mu_1= \beta + \phi$
$\alpha = \theta_1 + \mu_1$
$\delta = \mu_2 + \phi$
$\gamma = \mu_2 + \theta_2$
$\alpha = n \beta$
$\gamma = n \delta$

I have tried many things already such as equating expressions and trying to cancel unknowns etc but I think I am missing something. Any help would be greatly appreciated!

Reply 1

Eimmanuel

Study Forum Helper

Original post

by LionOnFire

Question Link: https://isaacphysics.org/questions/deriving_lensmaker

Hi, I am just stuck on part D as I am not sure how to derive the lensmaker's equation using the information given. From the previous parts and the information given in part D, I have found a few expressions which are relevant:
$R_1 \sin(\mu_1) = h$
$R_2 \sin(\mu_2) = h$
$u \tan(\theta_1) = h$
$v \tan(\theta_2) = h$
$\mu_2 = (\theta_1+\theta_2)/(n-1) - \mu_1$

I have also found these expressions form the previous parts (which I dont think are relevant but will still add):
$\mu_1= \beta + \phi$
$\alpha = \theta_1 + \mu_1$
$\delta = \mu_2 + \phi$
$\gamma = \mu_2 + \theta_2$
$\alpha = n \beta$
$\gamma = n \delta$

I have tried many things already such as equating expressions and trying to cancel unknowns etc but I think I am missing something. Any help would be greatly appreciated!

There are mainly 2 parts:
Consider the Figure 1 in part A, you have
$R_1 \sin(\mu_1) = h$ ---- (1a)
$u \tan(\theta_1) = h$ ------(1b)
and you need Snell’s law with the 2 following relationships that you have found
$\mu_1= \beta + \phi$ ------(1c)
$\alpha = \theta_1 + \mu_1$ ------(1d)
to connect the above 2 relationships (1a and 1b) using small angle approximation to form an equation in terms of h, u, R₁, n and ϕ, call this equation (1).

Next, consider Figure 2 and do the same “analysis” to find another equation in terms of h, v, R2, n ϕ, and call this equation (2).

Combine equation (1) and (2) into equation (3) in terms of R₁, R₂, u, v and n. You are 90% done.

The rest of the work, please read the question.

(edited 1 year ago)

Reply 2

LionOnFire

Original post

by Eimmanuel

There are mainly 2 parts:
Consider the Figure 1 in part A, you have
$R_1 \sin(\mu_1) = h$ ---- (1a)
$u \tan(\theta_1) = h$ ------(1b)
and you need Snell’s law with the 2 following relationships that you have found
$\mu_1= \beta + \phi$ ------(1c)
$\alpha = \theta_1 + \mu_1$ ------(1d)
to connect the above 2 relationships (1a and 1b) using small angle approximation to form an equation in terms of h, u, R₁, n and ϕ, call this equation (1).
Next, consider Figure 2 and do the same “analysis” to find another equation in terms of h, v, R2, n ϕ, and call this equation (2).
Combine equation (1) and (2) into equation (3) in terms of R₁, R₂, u, v and n. You are 90% done.
The rest of the work, please read the question.

Thank you, I got it! The main thing which stumped me previously was how to get rid of the sins and tans, is there any particular reason as to why you can just assume small angle approximations for them despite it not being said you can in the question?

Reply 3

Eimmanuel

Study Forum Helper

Original post

by LionOnFire

There is actually info in the question that is giving you hints on applying small angle approximation: in part A, it states “If α and β are small” and in part B, it states “If γ and δ are small” and part D is combining part A and part B by eliminating h and ϕ. In addition, hint 1 and 3 in D say small angle approximation.
If you have read any university physics text(s), most of the geometrical optics equations are derived using small angles approximation.

http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/paraxial.html
https://www.wikiwand.com/en/articles/Paraxial_approximation

Reply 4

LionOnFire

Original post

by Eimmanuel

There is actually info in the question that is giving you hints on applying small angle approximation: in part A, it states “If α and β are small” and in part B, it states “If γ and δ are small” and part D is combining part A and part B by eliminating h and ϕ. In addition, hint 1 and 3 in D say small angle approximation.
If you have read any university physics text(s), most of the geometrical optics equations are derived using small angles approximation.
http://hyperphysics.phy-astr.gsu.edu/hbase/geoopt/paraxial.html
https://www.wikiwand.com/en/articles/Paraxial_approximation

Oh ok, thanks for the help!

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