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review derivation: electric field wave equation: from time dependent to time independent

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
5 substitute LHS of expr 1 into expr 2
  1. 9499428242; locally 3994928:
    \(E( \vec{r},t) = E( \vec{r})\exp(i \omega t)\)
    \(\operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)} = \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}\)
  2. 9394939493; locally 3839493:
    \(\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)} = \frac{partial pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)}}{pdg_{1467}^{2}}\)
  1. 2029293929; locally 1029393:
    \(\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = \frac{partial pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}}{pdg_{1467}^{2}}\)
LHS diff is nabla**2*(pdg2718(pdg1467*pdg2321*pdg4621) - exp(pdg1467*pdg2321*pdg4621))*pdg6238(pdg9472) RHS diff is partial*pdg6197*pdg7940*(pdg2718(pdg1467*pdg2321*pdg4621) - exp(pdg1467*pdg2321*pdg4621))*pdg6238(pdg9472)/pdg1467**2 9499428242:
9394939493:
2029293929:
9499428242:
9394939493:
2029293929:
6 differentiate with respect to
  1. 2029293929; locally 1029393:
    \(\nabla^2 E( \vec{r})\exp(i \omega t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r})\exp(i \omega t)\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = \frac{partial pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}}{pdg_{1467}^{2}}\)
  1. 0003232242:
    \(t\)
    \(pdg_{1467}\)
  1. 4985825552; locally 2939392:
    \(\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = pdg_{2321} pdg_{4621} pdg_{6197} pdg_{7940} \frac{\partial}{\partial pdg_{1467}} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}\)
no check performed 2029293929:
4985825552:
2029293929:
4985825552:
2 declare initial expr
  1. 8572852424; locally 9393848:
    \(\vec{E} = E( \vec{r},t)\)
    \(pdg_{4326} = \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)}\)
no validation is available for declarations 8572852424:
8572852424:
3 declare guess solution
  1. 8494839423; locally 4758592:
    \(\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
    \(nabla^{2} pdg_{4326} = \frac{partial pdg_{4326} pdg_{6197} pdg_{7940}}{pdg_{1467}^{2}}\)
  1. 9499428242; locally 3994928:
    \(E( \vec{r},t) = E( \vec{r})\exp(i \omega t)\)
    \(\operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)} = \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}\)
no validation is available for declarations 8494839423:
9499428242:
8494839423:
9499428242:
4 substitute LHS of expr 1 into expr 2
  1. 8572852424; locally 9393848:
    \(\vec{E} = E( \vec{r},t)\)
    \(pdg_{4326} = \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)}\)
  2. 8494839423; locally 4758592:
    \(\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
    \(nabla^{2} pdg_{4326} = \frac{partial pdg_{4326} pdg_{6197} pdg_{7940}}{pdg_{1467}^{2}}\)
  1. 9394939493; locally 3839493:
    \(\nabla^2 E( \vec{r},t) = \mu_0 \epsilon_0 \frac{\partial^2}{\partial t^2} E( \vec{r},t)\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)} = \frac{partial pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472},pdg_{1467} \right)}}{pdg_{1467}^{2}}\)
valid 8572852424:
8494839423:
9394939493:
8572852424:
8494839423:
9394939493:
7 differentiate with respect to
  1. 4985825552; locally 2939392:
    \(\nabla^2 E( \vec{r})\exp(i \omega t) = i \omega \mu_0 \epsilon_0 \frac{\partial}{\partial t} E( \vec{r})\exp(i \omega t)\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = pdg_{2321} pdg_{4621} pdg_{6197} pdg_{7940} \frac{\partial}{\partial pdg_{1467}} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}\)
  1. 0003232242:
    \(t\)
    \(pdg_{1467}\)
  1. 1858578388; locally 4958573:
    \(\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = - pdg_{2321}^{2} pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}\)
no check performed 4985825552:
1858578388:
4985825552:
1858578388:
10 simplify
  1. 9485384858; locally 9495903:
    \(\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)\)
    \(nabla^{2} \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} = - \frac{pdg_{2321}^{2} \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}}{pdg_{4567}^{2}}\)
  1. 3485475729; locally 3949492:
    \(\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} = - \frac{pdg_{2321}^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}}{pdg_{4567}^{2}}\)
LHS diff is nabla**2*(pdg2718(pdg1467*pdg2321*pdg4621) - 1)*pdg6238(pdg9472) RHS diff is pdg2321**2*(1 - pdg2718(pdg1467*pdg2321*pdg4621))*pdg6238(pdg9472)/pdg4567**2 9485384858:
3485475729:
9485384858:
3485475729:
11 declare final expr
  1. 3485475729; locally 3949492:
    \(\nabla^2 E( \vec{r}) = - \frac{\omega^2}{c^2} E( \vec{r})\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} = - \frac{pdg_{2321}^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}}{pdg_{4567}^{2}}\)
no validation is available for declarations 3485475729:
3485475729:
9 substitute LHS of expr 1 into expr 2
  1. 1858578388; locally 4958573:
    \(\nabla^2 E( \vec{r})\exp(i \omega t) = - \omega^2 \mu_0 \epsilon_0 E( \vec{r})\exp(i \omega t)\)
    \(nabla^{2} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}} = - pdg_{2321}^{2} pdg_{6197} pdg_{7940} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} e^{pdg_{1467} pdg_{2321} pdg_{4621}}\)
  2. 4585828572; locally 4949582:
    \(\epsilon_0 \mu_0 = \frac{1}{c^2}\)
    \(pdg_{6197} pdg_{7940} = \frac{1}{pdg_{4567}^{2}}\)
  1. 9485384858; locally 9495903:
    \(\nabla^2 E( \vec{r})\exp(i \omega t) = - \frac{\omega^2}{c^2} E( \vec{r})\exp(i \omega t)\)
    \(nabla^{2} \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)} = - \frac{pdg_{2321}^{2} \operatorname{pdg_{2718}}{\left(pdg_{1467} pdg_{2321} pdg_{4621} \right)} \operatorname{pdg_{6238}}{\left(pdg_{9472} \right)}}{pdg_{4567}^{2}}\)
LHS diff is -nabla**2*pdg2718(pdg1467*pdg2321*pdg4621)*pdg6238(pdg9472) + pdg6197*pdg7940 RHS diff is (pdg2321**2*pdg2718(pdg1467*pdg2321*pdg4621)*pdg6238(pdg9472) + 1)/pdg4567**2 1858578388:
4585828572: dimensions are consistent
9485384858:
1858578388:
4585828572: N/A
9485384858:
1 declare initial expr
  1. 8494839423; locally 4758592:
    \(\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
    \(nabla^{2} pdg_{4326} = \frac{partial pdg_{4326} pdg_{6197} pdg_{7940}}{pdg_{1467}^{2}}\)
no validation is available for declarations 8494839423:
8494839423:
8 declare initial expr
  1. 4585828572; locally 4949582:
    \(\epsilon_0 \mu_0 = \frac{1}{c^2}\)
    \(pdg_{6197} pdg_{7940} = \frac{1}{pdg_{4567}^{2}}\)
no validation is available for declarations 4585828572: dimensions are consistent
4585828572: N/A
Physics Derivation Graph: Steps for electric field wave equation: from time dependent to time independent

Symbols for this derivation

See also all 212 symbols
symbol ID category latex scope dimension name value Used in derivations references
4326 variable \vec{E}
\(\vec{E}\)
complex dimensionless electric field 9
7940 constant \epsilon_0
\(\epsilon_0\)
real
  • length: -3
  • time: 2
  • mass: -1
  • charge: 2
vacuum permittivity, permittivity of free space or electric constant or the distributed capacitance of the vacuum 8.8541878128E-{12}   F/m
14
1467 variable t
\(t\)
['real']
  • time: 1
time 115
2718 constant \exp
\(\exp\)
['real']
e 2.71828   unitless
8
6238 variable E
\(E\)
real dimensionless electric field 20
2321 variable \omega
\(\omega\)
['real']
  • time: -1
angular frequency 15
4621 variable i
\(i\)
['imaginary']
imaginary unit 74
6197 constant \mu_0
\(\mu_0\)
real
  • length: 1
  • mass: 1
  • charge: -2
vacuum permeability, permeability of free space, permeability of vacuum, or magnetic constant 1.25663706212E^{-6}   N/A^2
8
4567 constant c
\(c\)
['real']
  • length: 1
  • time: -1
speed of light in vacuum 299792458   meters/second
32
9472 variable \vec{r}
\(\vec{r}\)
real
  • length: 1
radius vector
  • str_note
24
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