## 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: error for dim with 9999999981
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: error for dim with 9999999981
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

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
6238 variable E
$$E$$
real dimensionless electric field
20
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
3935 variable \rho
$$\rho$$
real
• length: -3
• mass: 1
density
7
2069 variable \vec{H}
$$\vec{H}$$
real dimensionless magnetic field
3
MESSAGES:
• saved to file
• in step 4655746: Dot(nabla, pdg6238) cannot be interpreted correctly