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

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

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
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:
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:
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:
8495187962:
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:
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:
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:
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: no LHS/RHS split
2575937347:
7696214507:
1310571337: N/A
2575937347:
7696214507:
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:
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:
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: no LHS/RHS split
8945218208: N/A
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:
3417126140:
8495187962:
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:
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:
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: no LHS/RHS split
1310571337: no LHS/RHS split
8945218208: N/A
1310571337: N/A
Physics Derivation Graph: Steps for optics: Law of refraction to Brewster's angle

Symbols for this derivation

See also all 215 symbols
symbol ID category latex scope dimension name value Used in derivations references
7545 variable \theta_2
\(\theta_2\)
real dimensionless angle 2
1464 variable x
\(x\)
['real'] dimensionless 140
2941 variable n_1
\(n_1\)
['real'] dimensionless index of refraction for material 1 9
3509 variable \theta_1
\(\theta_1\)
real dimensionless angle 2
2243 variable \theta_{\rm refracted}
\(\theta_{\rm refracted}\)
real dimensionless refracted angle
  • str_note
2
4928 variable \theta_{\rm Brewster}
\(\theta_{\rm Brewster}\)
['real'] dimensionless Brewster's angle 16
1958 variable n_2
\(n_2\)
real dimensionless index of refraction for material 2 8
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