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review derivation: total electrical resistance for circuit with two resistors in parallel

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
1 change two variables in expr
  1. 4087145886; locally 8922008:
    \(V = I R\)
    \(pdg_{6599} = pdg_{4501} pdg_{6458}\)
  1. 2867848403:
    \(I\)
    \(pdg_{4501}\)
  2. 1323602089:
    \(I_1\)
    \(pdg_{3978}\)
  3. 7191277455:
    \(R\)
    \(pdg_{6458}\)
  4. 5258419993:
    \(R_1\)
    \(pdg_{8697}\)
  1. 4128500715; locally 8148802:
    \(V = I_1 R_1\)
    \(pdg_{6599} = pdg_{3978} pdg_{8697}\)
valid 4087145886:
4128500715:
4087145886:
4128500715:
2 change two variables in expr
  1. 4087145886; locally 8922008:
    \(V = I R\)
    \(pdg_{6599} = pdg_{4501} pdg_{6458}\)
  1. 5585739998:
    \(I\)
    \(pdg_{4501}\)
  2. 5463275819:
    \(I_2\)
    \(pdg_{4856}\)
  3. 1377431959:
    \(R\)
    \(pdg_{6458}\)
  4. 9746066299:
    \(R_2\)
    \(pdg_{3461}\)
  1. 9243879541; locally 6313158:
    \(V = I_2 R_2\)
    \(pdg_{6599} = pdg_{3461} pdg_{4856}\)
valid 4087145886:
9243879541:
4087145886:
9243879541:
3 divide both sides by
  1. 9243879541; locally 6313158:
    \(V = I_2 R_2\)
    \(pdg_{6599} = pdg_{3461} pdg_{4856}\)
  1. 3031116098:
    \(R_2\)
    \(pdg_{3461}\)
  1. 7002609475; locally 4037937:
    \(\frac{V}{R_2} = I_2\)
    \(\frac{pdg_{6599}}{pdg_{3461}} = pdg_{4856}\)
valid 9243879541:
7002609475:
9243879541:
7002609475:
4 divide both sides by
  1. 4128500715; locally 8148802:
    \(V = I_1 R_1\)
    \(pdg_{6599} = pdg_{3978} pdg_{8697}\)
  1. 5181421075:
    \(R_1\)
    \(pdg_{8697}\)
  1. 2051901211; locally 5168370:
    \(\frac{V}{R_1} = I_1\)
    \(\frac{pdg_{6599}}{pdg_{8697}} = pdg_{3978}\)
valid 4128500715:
2051901211:
4128500715:
2051901211:
5 declare initial expr
  1. 6753224061; locally 3843569:
    \(I_{\rm total} = I_1 + I_2\)
    \(pdg_{9647} = pdg_{3978} + pdg_{4856}\)
no validation is available for declarations 6753224061:
6753224061:
current flows through both resistors
6 change two variables in expr
  1. 4087145886; locally 8922008:
    \(V = I R\)
    \(pdg_{6599} = pdg_{4501} pdg_{6458}\)
  1. 9053099840:
    \(I\)
    \(pdg_{4501}\)
  2. 7798615279:
    \(I_{\rm total}\)
    \(pdg_{9647}\)
  3. 1100332145:
    \(R\)
    \(pdg_{6458}\)
  4. 7410124465:
    \(R_{\rm total}\)
    \(pdg_{1908}\)
  1. 2271186630; locally 8002723:
    \(V = I_{\rm total} R_{\rm total}\)
    \(pdg_{6599} = pdg_{1908} pdg_{9647}\)
valid 4087145886:
2271186630:
4087145886:
2271186630:
7 divide both sides by
  1. 2271186630; locally 8002723:
    \(V = I_{\rm total} R_{\rm total}\)
    \(pdg_{6599} = pdg_{1908} pdg_{9647}\)
  1. 6546594355:
    \(R_{\rm total}\)
    \(pdg_{1908}\)
  1. 2809345867; locally 1708642:
    \(\frac{V}{R_{\rm total}} = I_{\rm total}\)
    \(\frac{pdg_{6599}}{pdg_{1908}} = pdg_{9647}\)
valid 2271186630:
2809345867:
2271186630:
2809345867:
8 substitute LHS of three expressions into expr
  1. 2809345867; locally 1708642:
    \(\frac{V}{R_{\rm total}} = I_{\rm total}\)
    \(\frac{pdg_{6599}}{pdg_{1908}} = pdg_{9647}\)
  2. 2051901211; locally 5168370:
    \(\frac{V}{R_1} = I_1\)
    \(\frac{pdg_{6599}}{pdg_{8697}} = pdg_{3978}\)
  3. 7002609475; locally 4037937:
    \(\frac{V}{R_2} = I_2\)
    \(\frac{pdg_{6599}}{pdg_{3461}} = pdg_{4856}\)
  4. 6753224061; locally 3843569:
    \(I_{\rm total} = I_1 + I_2\)
    \(pdg_{9647} = pdg_{3978} + pdg_{4856}\)
  1. 4866160902; locally 6759349:
    \(\frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2}\)
    \(\frac{pdg_{6599}}{pdg_{1908}} = \frac{pdg_{6599}}{pdg_{8697}} + \frac{pdg_{6599}}{pdg_{3461}}\)
no check performed 2809345867:
2051901211:
7002609475:
6753224061:
4866160902:
2809345867:
2051901211:
7002609475:
6753224061:
4866160902:
9 divide both sides by
  1. 4866160902; locally 6759349:
    \(\frac{V}{R_{\rm total}} = \frac{V}{R_1} + \frac{V}{R_2}\)
    \(\frac{pdg_{6599}}{pdg_{1908}} = \frac{pdg_{6599}}{pdg_{8697}} + \frac{pdg_{6599}}{pdg_{3461}}\)
  1. 3433441359:
    \(V\)
    \(pdg_{6599}\)
  1. 1457415749; locally 8713399:
    \(\frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2}\)
    \(\frac{1}{pdg_{1908}} = \frac{1}{pdg_{8697}} + \frac{1}{pdg_{3461}}\)
valid 4866160902:
1457415749: dimensions are consistent
4866160902:
1457415749: N/A
10 declare final expr
  1. 1457415749; locally 8713399:
    \(\frac{1}{R_{\rm total}} = \frac{1}{R_1} + \frac{1}{R_2}\)
    \(\frac{1}{pdg_{1908}} = \frac{1}{pdg_{8697}} + \frac{1}{pdg_{3461}}\)
no validation is available for declarations 1457415749: dimensions are consistent
1457415749: N/A
Physics Derivation Graph: Steps for total electrical resistance for circuit with two resistors in parallel

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
3978 variable I_1
\(I_1\)
real
  • time: -1
  • charge: 1
electric current 4
1908 variable R_{\rm total}
\(R_{\rm total}\)
real
  • length: 2
  • time: -1
  • mass: 1
  • charge: -2
electrical resistance 9
3461 variable R_2
\(R_2\)
real
  • length: 2
  • time: -1
  • mass: 1
  • charge: -2
electrical resistance 10
9647 variable I_{\rm total}
\(I_{\rm total}\)
real
  • time: -1
  • charge: 1
electric current 4
4501 variable I
\(I\)
real
  • time: -1
  • charge: 1
electric current 9
6599 variable V
\(V\)
real
  • length: 2
  • time: -2
  • mass: 1
  • charge: -1
voltage 11
6458 variable R
\(R\)
real
  • length: 2
  • time: -1
  • mass: 1
  • charge: -2
electrical resistance 6
4856 variable I_2
\(I_2\)
real
  • time: -1
  • charge: 1
electric current 4
8697 variable R_1
\(R_1\)
real
  • length: 2
  • time: -1
  • mass: 1
  • charge: -2
electrical resistance 10
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