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review derivation: Maxwell equations to electric field wave equation

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Index Inference Rule Input latex Feeds latex Output latex step validity dimension check unit check notes
3 partially differentiate with respect to
  1. 1314864131; locally 1199299:
    \(\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}\)
    \(nabla \times pdg_{2069} = pdg_{7940} \frac{d}{d pdg_{1467}} pdg_{4326}\)
  1. 0005626421:
    \(t\)
    \(pdg_{1467}\)
  1. 1314464131; locally 4642245:
    \(\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
    \(pdg_{1467}\)
Nothing to split 1314864131:
1314464131:
1314864131:
1314464131:
5 substitute LHS of expr 1 into expr 2
  1. 9291999979; locally 2392932:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}\)
    \(nabla^{2} pdg_{4326} = - nabla pdg_{6197} \frac{d}{d pdg_{1467}} pdg_{2069}\)
  2. 1314464131; locally 4642245:
    \(\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t} = \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
    \(pdg_{1467}\)
  1. 3947269979; locally 2962831:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
    \(pdg_{1467}\)
Nothing to split 9291999979:
1314464131:
3947269979:
9291999979:
1314464131:
3947269979:
1 declare initial expr
  1. 9991999979; locally 4757562:
    \(\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}\)
    \(- nabla \times pdg_{6238} = - pdg_{6197} \frac{d}{d pdg_{1467}} pdg_{2069}\)
no validation is available for declarations 9991999979:
9991999979:
1.3 declare initial expr
  1. 1314864131; locally 1199299:
    \(\vec{ \nabla} \times \vec{H} = \epsilon_0 \frac{\partial }{\partial t}\vec{E}\)
    \(nabla \times pdg_{2069} = pdg_{7940} \frac{d}{d pdg_{1467}} pdg_{4326}\)
no validation is available for declarations 1314864131:
1314864131:
1.6 declare initial expr
  1. 9999999981; locally 4857731:
    \(\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0\)
    \(nabla pdg_{4326} = \frac{pdg_{3935}}{pdg_{7940}}\)
no validation is available for declarations 9999999981: failed
 9999999981: N/A
11 substitute LHS of expr 1 into expr 2
  1. 1636453295; locally 3738373:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}\)
    \(nabla \times \left(nabla \times pdg_{4326}\right) = - nabla^{2} pdg_{4326}\)
  2. 3947269979; locally 2962831:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
    \(pdg_{1467}\)
  1. 8494839423; locally 4758592:
    \(\nabla^2 \vec{E} = \mu_0 \epsilon_0 \frac{\partial^2 \vec{E}}{\partial t^2}\)
    \(nabla^{2} pdg_{4326} = \frac{partial pdg_{4326} pdg_{6197} pdg_{7940}}{pdg_{1467}^{2}}\)
Nothing to split 1636453295:
3947269979:
8494839423:
1636453295:
3947269979:
8494839423:
7 declare assumption
  1. 9919999981; locally 3984852:
    \(\rho = 0\)
    \(pdg_{3935} = 0\)
no validation is available for declarations 9919999981: error for dim with 9919999981
 9919999981: N/A
9 declare identity
  1. 7575859295; locally 1939485:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(pdg_{6238} \times \left(nabla \times nabla\right) = \operatorname{nabla}{\left(- nabla^{2} pdg_{6238} + nabla \cdot pdg_{6238} \right)}\)
no validation is available for declarations 7575859295:
7575859295:
12 declare final 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:
4 take curl of both sides
  1. 9991999979; locally 4757562:
    \(\vec{ \nabla} \times \vec{E} = -\mu_0\frac{\partial \vec{H}}{\partial t}\)
    \(- nabla \times pdg_{6238} = - pdg_{6197} \frac{d}{d pdg_{1467}} pdg_{2069}\)
  1. 9291999979; locally 2392932:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = -\mu_0\vec{ \nabla} \times \frac{\partial \vec{H}}{\partial t}\)
    \(nabla^{2} pdg_{4326} = - nabla pdg_{6197} \frac{d}{d pdg_{1467}} pdg_{2069}\)
no check performed 9991999979:
9291999979:
9991999979:
9291999979:
10 substitute LHS of expr 1 into expr 2
  1. 7575859295; locally 1939485:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = \vec{ \nabla}( \vec{ \nabla} \cdot \vec{E} - \nabla^2 \vec{E})\)
    \(pdg_{6238} \times \left(nabla \times nabla\right) = \operatorname{nabla}{\left(- nabla^{2} pdg_{6238} + nabla \cdot pdg_{6238} \right)}\)
  2. 7466829492; locally 2837471:
    \(\vec{ \nabla} \cdot \vec{E} = 0\)
    \(nabla \cdot pdg_{6238} = 0\)
  1. 1636453295; locally 3738373:
    \(\vec{ \nabla} \times \vec{ \nabla} \times \vec{E} = - \nabla^2 \vec{E}\)
    \(nabla \times \left(nabla \times pdg_{4326}\right) = - nabla^{2} pdg_{4326}\)
failed 7575859295:
7466829492: error for dim with 7466829492
1636453295:
7575859295:
7466829492: N/A
1636453295:
8 substitute LHS of expr 1 into expr 2
  1. 9999999981; locally 4857731:
    \(\vec{ \nabla} \cdot \vec{E} = \rho/\epsilon_0\)
    \(nabla pdg_{4326} = \frac{pdg_{3935}}{pdg_{7940}}\)
  2. 9919999981; locally 3984852:
    \(\rho = 0\)
    \(pdg_{3935} = 0\)
  1. 7466829492; locally 2837471:
    \(\vec{ \nabla} \cdot \vec{E} = 0\)
    \(nabla \cdot pdg_{6238} = 0\)
LHS diff is pdg3935 - Dot(nabla, pdg6238) RHS diff is 0 9999999981: failed
9919999981: error for dim with 9919999981
7466829492: error for dim with 7466829492
 9999999981: N/A
9919999981: N/A
7466829492: N/A
Physics Derivation Graph: Steps for Maxwell equations to electric field wave equation

Symbols for this derivation

See also all 227 symbols
symbol ID category latex scope dimension name value Used in derivations references
7940 constant \epsilon_0
\(\epsilon_0\)		 real
  • electric charge: 2
  • length: -3
  • mass: -1
  • time: 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 121
2069 variable \vec{H}
\(\vec{H}\)		 real dimensionless magnetic field 3
4326 variable \vec{E}
\(\vec{E}\)		 complex dimensionless electric field 9
6238 variable E
\(E\)		 real dimensionless electric field 20
6197 constant \mu_0
\(\mu_0\)		 real
  • electric charge: -2
  • length: 1
  • mass: 1
 vacuum permeability, permeability of free space, permeability of vacuum, or magnetic constant 1.25663706212E^{-6}   N/A^2
 8
3935 variable \rho
\(\rho\)		 real
  • length: -3
  • mass: 1
 density 7
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