Physics Derivation Graph navigation Sign in

review derivation: optics: Law of refraction to Brewster's angle

This page contains three views of the steps in the derivation: d3js, graphviz PNG, and a table.


Hold the mouse over a node to highlight that node and its neighbors. You can zoom in/out. You can pan the image. You can move nodes by clicking and dragging.

Notes for this derivation:
\cite{2001_HRW}; see figure 34-27 on page 824

Options
Alternate views of this derivation:
Edit this content:    

To edit a step, click on the number in the "Index" column in the table below

Clicking on the step index will take you to the page where you can edit that step.

Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
1 declare initial expr
  1. 8945218208; locally 5563180:
    \(\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}\)
    \(pdg_{4928}\)
no validation is available for declarations 8945218208:
8945218208:
2 subtract X from both sides
  1. 8945218208; locally 5563180:
    \(\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}\)
    \(pdg_{4928}\)
  1. 9025853427:
    \(\theta_{\rm Brewster}\)
    \(pdg_{4928}\)
  1. 1310571337; locally 3893026:
    \(\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}\)
    \(pdg_{4928}\)
Nothing to split 8945218208:
1310571337:
8945218208:
1310571337:
3 declare initial expr
  1. 6450985774; locally 9932375:
    \(n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )\)
    \(pdg_{2941} \sin{\left(pdg_{3509} \right)} = pdg_{1958} \sin{\left(pdg_{7545} \right)}\)
no validation is available for declarations 6450985774:
6450985774:
4 change two variables in expr
  1. 6450985774; locally 9932375:
    \(n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )\)
    \(pdg_{2941} \sin{\left(pdg_{3509} \right)} = pdg_{1958} \sin{\left(pdg_{7545} \right)}\)
  1. 7154592211:
    \(\theta_2\)
    \(pdg_{7545}\)
  2. 6353701615:
    \(\theta_{\rm refracted}\)
    \(pdg_{2243}\)
  3. 2773628333:
    \(\theta_1\)
    \(pdg_{3509}\)
  4. 9029795851:
    \(\theta_{\rm Brewster}\)
    \(pdg_{4928}\)
  1. 2575937347; locally 4176694:
    \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )\)
    \(pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \sin{\left(pdg_{2243} \right)}\)
valid 6450985774:
2575937347:
6450985774:
2575937347:
5 substitute LHS of expr 1 into expr 2
  1. 1310571337; locally 3893026:
    \(\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}\)
    \(pdg_{4928}\)
  2. 2575937347; locally 4176694:
    \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )\)
    \(pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \sin{\left(pdg_{2243} \right)}\)
  1. 7696214507; locally 4962698:
    \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )\)
    \(pdg_{2941} \sin{\left(pdg_{4928} \right)} = - pdg_{1958} \sin{\left(pdg_{4928} - 90 \right)}\)
Nothing to split 1310571337:
2575937347:
7696214507:
1310571337:
2575937347:
7696214507:
6 declare identity
  1. 8588429722; locally 3940135:
    \(\sin( 90^{\circ} - x ) = \cos( x )\)
    \(- \sin{\left(pdg_{1464} - 90 \right)} = \cos{\left(pdg_{1464} \right)}\)
no validation is available for declarations 8588429722:
8588429722:
7 change variable X to Y
  1. 8588429722; locally 3940135:
    \(\sin( 90^{\circ} - x ) = \cos( x )\)
    \(- \sin{\left(pdg_{1464} - 90 \right)} = \cos{\left(pdg_{1464} \right)}\)
  1. 7375348852:
    \(\theta_{\rm Brewster}\)
    \(pdg_{4928}\)
  2. 1512581563:
    \(x\)
    \(pdg_{1464}\)
  1. 6831637424; locally 7426234:
    \(\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )\)
    \(pdg_{4928}\)
Nothing to split 8588429722:
6831637424:
8588429722:
6831637424:
8 substitute LHS of expr 1 into expr 2
  1. 6831637424; locally 7426234:
    \(\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )\)
    \(pdg_{4928}\)
  2. 7696214507; locally 4962698:
    \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )\)
    \(pdg_{2941} \sin{\left(pdg_{4928} \right)} = - pdg_{1958} \sin{\left(pdg_{4928} - 90 \right)}\)
  1. 3061811650; locally 9701820:
    \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )\)
    \(pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \cos{\left(pdg_{4928} \right)}\)
Nothing to split 6831637424:
7696214507:
3061811650:
6831637424:
7696214507:
3061811650:
9 divide both sides by
  1. 3061811650; locally 9701820:
    \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )\)
    \(pdg_{2941} \sin{\left(pdg_{4928} \right)} = pdg_{1958} \cos{\left(pdg_{4928} \right)}\)
  1. 7857757625:
    \(n_1\)
    \(pdg_{2941}\)
  1. 9756089533; locally 9314305:
    \(\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )\)
    \(\sin{\left(pdg_{4928} \right)} = \frac{pdg_{1958} \cos{\left(pdg_{4928} \right)}}{pdg_{2941}}\)
valid 3061811650:
9756089533:
3061811650:
9756089533:
10 divide both sides by
  1. 9756089533; locally 9314305:
    \(\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )\)
    \(\sin{\left(pdg_{4928} \right)} = \frac{pdg_{1958} \cos{\left(pdg_{4928} \right)}}{pdg_{2941}}\)
  1. 5632428182:
    \(\cos( \theta_{\rm Brewster} )\)
    \(\cos{\left(pdg_{4928} \right)}\)
  1. 2768857871; locally 8585856:
    \(\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}\)
    \(\frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}} = \frac{pdg_{1958}}{pdg_{2941}}\)
valid 9756089533:
2768857871:
9756089533:
2768857871:
11 declare identity
  1. 4968680693; locally 2621708:
    \(\tan( x ) = \frac{ \sin( x )}{\cos( x )}\)
    \(\tan{\left(pdg_{1464} \right)} = \frac{\sin{\left(pdg_{1464} \right)}}{\cos{\left(pdg_{1464} \right)}}\)
no validation is available for declarations 4968680693:
4968680693:
12 change variable X to Y
  1. 4968680693; locally 2621708:
    \(\tan( x ) = \frac{ \sin( x )}{\cos( x )}\)
    \(\tan{\left(pdg_{1464} \right)} = \frac{\sin{\left(pdg_{1464} \right)}}{\cos{\left(pdg_{1464} \right)}}\)
  1. 7321695558:
    \(\theta_{\rm Brewster}\)
    \(pdg_{4928}\)
  2. 9906920183:
    \(x\)
    \(pdg_{1464}\)
  1. 4501377629; locally 1898054:
    \(\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\)
    \(\tan{\left(pdg_{4928} \right)} = \frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}}\)
LHS diff is tan(pdg1464) - tan(pdg4928) RHS diff is tan(pdg1464) - tan(pdg4928) 4968680693:
4501377629:
4968680693:
4501377629:
13 substitute LHS of expr 1 into expr 2
  1. 4501377629; locally 1898054:
    \(\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\)
    \(\tan{\left(pdg_{4928} \right)} = \frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}}\)
  2. 2768857871; locally 8585856:
    \(\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}\)
    \(\frac{\sin{\left(pdg_{4928} \right)}}{\cos{\left(pdg_{4928} \right)}} = \frac{pdg_{1958}}{pdg_{2941}}\)
  1. 3417126140; locally 5179630:
    \(\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }\)
    \(\tan{\left(pdg_{4928} \right)} = \frac{pdg_{1958}}{pdg_{2941}}\)
valid 4501377629:
2768857871:
3417126140:
4501377629:
2768857871:
3417126140:
14 apply function to both sides of expression
  1. 3417126140; locally 5179630:
    \(\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }\)
    \(\tan{\left(pdg_{4928} \right)} = \frac{pdg_{1958}}{pdg_{2941}}\)
  1. 5453995431:
    \(\arctan{ x }\)
    \(\operatorname{atan}{\left(pdg_{1464} \right)}\)
  2. 6023986360:
    \(x\)
    \(pdg_{1464}\)
  1. 8495187962; locally 8186016:
    \(\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }\)
    \(pdg_{4928} = \operatorname{atan}{\left(\frac{pdg_{2941}}{pdg_{1958}} \right)}\)
no check performed 3417126140:
8495187962: error for dim with 8495187962
3417126140:
8495187962: N/A
15 declare final expr
  1. 8495187962; locally 8186016:
    \(\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }\)
    \(pdg_{4928} = \operatorname{atan}{\left(\frac{pdg_{2941}}{pdg_{1958}} \right)}\)
no validation is available for declarations 8495187962: error for dim with 8495187962
8495187962: N/A
Physics Derivation Graph: Steps for optics: Law of refraction to Brewster's angle

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
1464 variable x
\(x\)
['real']
140
1958 variable n_2
\(n_2\)
real
index of refraction for material 2 8
7545 variable \theta_2
\(\theta_2\)
real
angle 2
3509 variable \theta_1
\(\theta_1\)
real
angle 2
4928 variable \theta_{\rm Brewster}
\(\theta_{\rm Brewster}\)
['real']
Brewster's angle 16
2941 variable n_1
\(n_1\)
['real']
index of refraction for material 1 9
2243 variable \theta_{\rm refracted}
\(\theta_{\rm refracted}\)
real
refracted angle
  • str_note
2
MESSAGE: