Review Derivation: Physics Derivation Graph

MESSAGE:

pdg_app/to_review_derivation sympy_validate_step.validate_step: AssertionError 0008072315

Return to navigation page or list derivations

Review optics: Law of refraction to Brewster's angle

abstract: \cite{2001_HRW}; see figure 34-27 on page 824
reference:

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			8945218208 $\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}$	no validation is available for declarations
2	0000111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	8945218208 $\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}$	9025853427 $\theta_{\rm Brewster}$	1310571337 $\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}$	Not evaluated due to missing term in SymPy
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			6450985774 $n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )$	no validation is available for declarations
4	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	6450985774 $n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )$	9029795851 $\theta_{\rm Brewster}$ 2773628333 $\theta_1$ 6353701615 $\theta_{\rm refracted}$ 7154592211 $\theta_2$	2575937347 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )$	LHS diff is pdg0002941(sin(pdg0003509) - sin(pdg0004928)) RHS diff is pdg0001958(-sin(pdg0002243) + sin(pdg0007545))
5	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2575937347 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )$ 1310571337 $\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}$		7696214507 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )$	Not evaluated due to missing term in SymPy
6	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			8588429722 $\sin( 90^{\circ} - x ) = \cos( x )$	no validation is available for declarations
7	0000111886: change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	8588429722 $\sin( 90^{\circ} - x ) = \cos( x )$	1512581563 $x$ 7375348852 $\theta_{\rm Brewster}$	6831637424 $\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )$	Not evaluated due to missing term in SymPy
8	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	7696214507 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )$ 6831637424 $\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )$		3061811650 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )$	Not evaluated due to missing term in SymPy
9	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	3061811650 $n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )$	7857757625 $n_1$	9756089533 $\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )$	valid
10	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	9756089533 $\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )$	5632428182 $\cos( \theta_{\rm Brewster} )$	2768857871 $\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}$	valid
11	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			4968680693 $\tan( x ) = \frac{ \sin( x )}{\cos( x )}$	no validation is available for declarations
12	0000111886: change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	4968680693 $\tan( x ) = \frac{ \sin( x )}{\cos( x )}$	9906920183 $x$ 7321695558 $\theta_{\rm Brewster}$	4501377629 $\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}$	valid
13	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2768857871 $\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}$ 4501377629 $\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}$		3417126140 $\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }$	valid
14	0000111490: apply function to both sides of expression number of inputs: 1; feeds: 2; outputs: 1 Apply function $#1$ with argument $#2$ to Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}	3417126140 $\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }$	6023986360 $x$ 5453995431 $\arctan{ x }$	8495187962 $\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }$
15	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	8495187962 $\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }$			no validation is available for declarations

Symbols used in optics: Law of refraction to Brewster's angle

0000003509: $ \theta_1 $
0000007545: $ \theta_2 $
0000004928: $ \theta_{\rm Brewster} $
0000002243: $ \theta_{\rm refracted} $
0000002941: $ n_1 $
0000001958: $ n_2 $
0000001464: $ x $

$Steps and expressions for optics: Law of refraction to Brewster's angle$

d3js visualization of steps and expressions in optics: Law of refraction to Brewster's angle

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.

Actions: Edit Derivation

You must be logged in to edit the steps of derivation, edit the abstract, or rename the derivation.

Generate Tex file or PDF file

Delete Derivation and all associated steps

You must be logged in to delete the derivation.

timing of Neo4j queries:

0.027 seconds for pdg_app/to_review_derivation 4079eb3f-9c1a-45b6-8ed4-12d77a440eb6
0.034 seconds for compute/get_dict_of_steps_in_derivation: steps_in_this_derivationb90ba931-15fb-42a9-b4b7-9122983e8fb8
0.006 seconds for compute/input_feed_output_infrule_for_step: get_inference_rule_connected_to_step_IDb90ba931-15fb-42a9-b4b7-9122983e8fb8
0.014 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type HAS_INPUTb90ba931-15fb-42a9-b4b7-9122983e8fb8
0.01 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_FEEDb90ba931-15fb-42a9-b4b7-9122983e8fb8
0.006 seconds for compute/input_feed_output_infrule_for_step: get_expressions_from_step_id_and_expr_type, HAS_OUTPUTb90ba931-15fb-42a9-b4b7-9122983e8fb8
0.004 seconds for compute/get_dict_of_steps_in_derivation: get_sequence_index_for_stepb90ba931-15fb-42a9-b4b7-9122983e8fb8
0.007 seconds for pdg_app/ 4079eb3f-9c1a-45b6-8ed4-12d77a440eb6

step	inference rule	input	feed	output	step validity (as per SymPy)
1	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			8945218208 \(\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}\)	no validation is available for declarations
2	0000111282: subtract X from both sides number of inputs: 1; feeds: 1; outputs: 1 Subtract $#1$ from both sides of Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	8945218208 \(\theta_{\rm Brewster} + \theta_{\rm refracted} = 90^{\circ}\)	9025853427 \(\theta_{\rm Brewster}\)	1310571337 \(\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}\)	Not evaluated due to missing term in SymPy
3	0000111981: declare initial expression number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an initial equation.			6450985774 \(n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )\)	no validation is available for declarations
4	0000111984: change two variables in expression number of inputs: 1; feeds: 4; outputs: 1 Change variable $#1$ to $#2$ and $#3$ to $#4$ in Eq.~\ref{eq:#5}; yields Eq.~\ref{eq:#6}.	6450985774 \(n_1 \sin( \theta_1 ) = n_2 \sin( \theta_2 )\)	9029795851 \(\theta_{\rm Brewster}\) 2773628333 \(\theta_1\) 6353701615 \(\theta_{\rm refracted}\) 7154592211 \(\theta_2\)	2575937347 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )\)	LHS diff is pdg0002941(sin(pdg0003509) - sin(pdg0004928)) RHS diff is pdg0001958(-sin(pdg0002243) + sin(pdg0007545))
5	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2575937347 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( \theta_{\rm refracted} )\) 1310571337 \(\theta_{\rm refracted} = 90^{\circ} - \theta_{\rm Brewster}\)		7696214507 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )\)	Not evaluated due to missing term in SymPy
6	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			8588429722 \(\sin( 90^{\circ} - x ) = \cos( x )\)	no validation is available for declarations
7	0000111886: change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	8588429722 \(\sin( 90^{\circ} - x ) = \cos( x )\)	1512581563 \(x\) 7375348852 \(\theta_{\rm Brewster}\)	6831637424 \(\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )\)	Not evaluated due to missing term in SymPy
8	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	7696214507 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \sin( 90^{\circ} - \theta_{\rm Brewster} )\) 6831637424 \(\sin( 90^{\circ} - \theta_{\rm Brewster} ) = \cos( \theta_{\rm Brewster} )\)		3061811650 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )\)	Not evaluated due to missing term in SymPy
9	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	3061811650 \(n_1 \sin( \theta_{\rm Brewster} ) = n_2 \cos( \theta_{\rm Brewster} )\)	7857757625 \(n_1\)	9756089533 \(\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )\)	valid
10	0000111975: divide both sides by number of inputs: 1; feeds: 1; outputs: 1 Divide both sides of Eq.~\ref{eq:#2} by $#1$; yields Eq.~\ref{eq:#3}.	9756089533 \(\sin( \theta_{\rm Brewster} ) = \frac{n_2}{n_1} \cos( \theta_{\rm Brewster} )\)	5632428182 \(\cos( \theta_{\rm Brewster} )\)	2768857871 \(\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}\)	valid
11	0000111299: declare identity number of inputs: 0; feeds: 0; outputs: 1 Eq.~\ref{eq:#1} is an identity.			4968680693 \(\tan( x ) = \frac{ \sin( x )}{\cos( x )}\)	no validation is available for declarations
12	0000111886: change variable X to Y number of inputs: 1; feeds: 2; outputs: 1 Change variable $#1$ to $#2$ in Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}.	4968680693 \(\tan( x ) = \frac{ \sin( x )}{\cos( x )}\)	9906920183 \(x\) 7321695558 \(\theta_{\rm Brewster}\)	4501377629 \(\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\)	valid
13	0000111556: substitute LHS of expr 1 into expr 2 number of inputs: 2; feeds: 0; outputs: 1 Substitute LHS of Eq.~\ref{eq:#1} into Eq.~\ref{eq:#2}; yields Eq.~\ref{eq:#3}.	2768857871 \(\frac{\sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )} = \frac{n_2}{n_1}\) 4501377629 \(\tan( \theta_{\rm Brewster} ) = \frac{ \sin( \theta_{\rm Brewster} )}{\cos( \theta_{\rm Brewster} )}\)		3417126140 \(\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }\)	valid
14	0000111490: apply function to both sides of expression number of inputs: 1; feeds: 2; outputs: 1 Apply function $#1$ with argument $#2$ to Eq.~\ref{eq:#3}; yields Eq.~\ref{eq:#4}	3417126140 \(\tan( \theta_{\rm Brewster} ) = \frac{ n_2 }{ n_1 }\)	6023986360 \(x\) 5453995431 \(\arctan{ x }\)	8495187962 \(\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }\)
15	0000111341: declare final expression number of inputs: 1; feeds: 0; outputs: 0 Eq.~\ref{eq:#1} is one of the final equations.	8495187962 \(\theta_{\rm Brewster} = \arctan{ \left( \frac{ n_1 }{ n_2 } \right) }\)			no validation is available for declarations