Premium Only Content
Exercise 2: Deriving the 1st Order and 3rd Order Optics Formulas
In this video, I derive the first-order and third-order optics formulas by approximating the refraction formula for light using first and third-order Taylor polynomials, respectively. Refraction refers to the change in the direction of light as it passes between different mediums due to a change in speed. Light naturally travels in a way that minimizes time, and this principle helps us derive formulas for how light refracts when entering a material. The distances light travels before and after passing through an optical lens can be calculated using the Law of Cosines, which leads to a complex equation. However, we simplify this equation using Taylor polynomial approximations.
In the first part of the video, I derive the first-order, or Gaussian optics, formula by approximating the cosine function in the distance equations with its first term, which is simply 1.
In the second part, I derive the third-order Gaussian optics formula by using the third-degree Taylor polynomial for cosine, which includes the first two non-zero terms. I also apply a given hint to approximate the inverse of the distances using the first two terms of their respective binomial series. Additionally, I use a second hint to approximate the angle with its sine, which introduces a term involving the height. After working through the algebra, I arrive at the third-order optics formula!
Timestamps:
- Exercise 2: Derive 1st Order and 3rd Order Optics Equations for refraction: 0:00
- Solution to (a): Approximating cosine with First degree Taylor Polynomial: 0:59
- Equation 3: Gaussian optics or 1st order formula: 8:26
- Solution to (b): Approximating cosine with Third degree Taylor Polynomial: 9:19
- Modifying lengths in form of Binomial series: 17:46
- 1st Hint: Approximate lengths using first 2 terms of Binomial series: 23:20
- Plugging in our approximations into Equation 1: 28:06
- 2nd Hint: Approximate ɸ with sin(ɸ) to include the height term: 43:48
- Equation 4: 3rd order optics formula: 46:37
Full video, notes, and playlists:
- Full video and playlist: https://www.youtube.com/playlist?list=PLai3U8-WIK0F76sIU8xm09oqBTq1mlry3
- HIVE notes: https://peakd.com/hive-128780/@mes/infinite-sequences-and-series-applications-of-taylor-polynomials
- Infinite Sequences and Series: https://www.youtube.com/playlist?list=PLai3U8-WIK0EXHAJ3vRg0T_kKEyPah1Lz .
------------------------------------------------------
Become a MES Super Fan! https://www.youtube.com/channel/UCUUBq1GPBvvGNz7dpgO14Ow/join
DONATE! ʕ •ᴥ•ʔ https://mes.fm/donate
SUBSCRIBE via EMAIL: https://mes.fm/subscribe
MES Links: https://mes.fm/links
MES Truth: https://mes.fm/truth
Official Website: https://MES.fm
Hive: https://peakd.com/@mes
Email me: contact@mes.fm
Free Calculators: https://mes.fm/calculators
BMI Calculator: https://bmicalculator.mes.fm
Grade Calculator: https://gradecalculator.mes.fm
Mortgage Calculator: https://mortgagecalculator.mes.fm
Percentage Calculator: https://percentagecalculator.mes.fm
Free Online Tools: https://mes.fm/tools
iPhone and Android Apps: https://mes.fm/mobile-apps
-
4:01:13
Math Easy Solutions
6 days ago $0.24 earnedMES Livestream 61: America's Last Election Predictive Programming
617 -
14:13
Scammer Payback
12 days agoTelling Scammers Their Address
49.2K67 -
LIVE
Barstool Gambling
6 hours agoBig Cat and Co Sweat Out the Week 10 Sunday Slate | Barstool Gambling Cave
1,128 watching -
2:49:36
The Jimmy Dore Show
2 days agoRumble Time Live w/ Jimmy Dore & Special Guests Roseanne Barr, Dr. Drew, Drea de Matteo & More!
499K605 -
17:17
DeVory Darkins
19 hours agoKamala Post-Election BOMBSHELL Exposes $1 BILLION Campaign DISASTER
19.6K108 -
19:52
Stephen Gardner
23 hours ago🔥HOLY CRAP! Trump just did the UNTHINKABLE!!
35K392 -
4:34:55
Pepkilla
6 hours agoBlackops Terminus Zombies Boat Glitch
115K6 -
5:50
CapEx
18 hours ago $22.92 earnedWhat the Coming & Inevitable Sovereign Debt Crisis Means for YOU | CapEx Insider
108K25 -
1:34:00
Tactical Advisor
7 hours agoAR15 Giveaway WINNER/Trump Winning | Vault Room Live Stream 008
64.2K25 -
5:41:10
Vigilant News Network
9 hours agoOfficials CAUGHT Changing Ballots in Arizona | The Daily Dose
122K62