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Question: Study Maxwell’s Equations

Question:Study Maxwell’s Equations

Steve Roper 180 Maple
vector-calculus physics
March 11 2020
7

I'm attempting to use Maple to study Maxwell's Equations, but as a newbie to Maple, I quickly became stuck :-)

For some context, this link shows how it is possible to accomplish this using Mathematica:

https://www.wolfram.com/mathematica/new-in-10/inactive-objects/study-maxwells-equations.html

This is how I have attempted the same in Maple:

Maxwell's Equations

NULL

Initialise

 

restart

with(Physics[Vectors])

[`&x`, `+`, `.`, ChangeBasis, ChangeCoordinates, Component, Curl, DirectionalDiff, Divergence, Gradient, Identify, Laplacian, Nabla, Norm, Setup, diff]

 (1.1)

Setup(mathematicalnotation = true)

[mathematicalnotation = true]

(1.2)

``

````

Maxwell's Equations

 

Maxwell_1 := Curl(E__field_(x, y, z, t)) = -(diff(B__flux_(x, y, z, t), t))

Physics:-Vectors:-Curl(E__field_(x, y, z, t)) = -(diff(B__flux_(x, y, z, t), t))

(2.1)

Maxwell_2 := Curl(H__field_(x, y, z, t)) = diff(D__flux_(x, y, z, t), t)

Physics:-Vectors:-Curl(H__field_(x, y, z, t)) = diff(D__flux_(x, y, z, t), t)

(2.2)

Maxwell_3 := Divergence(D__flux_(x, y, z, t)) = 0

Physics:-Vectors:-Divergence(D__flux_(x, y, z, t)) = 0

(2.3)

Maxwell_4 := Divergence(B__flux_(x, y, z, t)) = 0

Physics:-Vectors:-Divergence(B__flux_(x, y, z, t)) = 0

(2.4)

``

``

Constitutive Relations

 

Eq_1 := D__flux_(x, y, z, t) = epsilon*E__field_(x, y, z, t)

D__flux_(x, y, z, t) = varepsilon*E__field_(x, y, z, t)

(3.1)

Eq_2 := B__flux_(x, y, z, t) = mu*H__field_(x, y, z, t)

B__flux_(x, y, z, t) = mu*H__field_(x, y, z, t)

(3.2)

``

``

Solution

 

We need to get the Curl of H, to take the Curl of both side of Maxwell_1:

Curl(Maxwell_1)

Physics:-Vectors:-Curl(Physics:-Vectors:-Curl(E__field_(x, y, z, t))) = -Physics:-Vectors:-Curl(diff(B__flux_(x, y, z, t), t))

(4.1)

Now substitute B for H:

subs(Eq_2, %)

Physics:-Vectors:-Curl(Physics:-Vectors:-Curl(E__field_(x, y, z, t))) = -Physics:-Vectors:-Curl(diff(mu*H__field_(x, y, z, t), t))

(4.2)

OK, we manage to get Curl of H, so now we need to substitute the Curl of H with an expression in D.

subs(Maxwell_2, %)

Physics:-Vectors:-Curl(Physics:-Vectors:-Curl(E__field_(x, y, z, t))) = -mu*Physics:-Vectors:-Curl(diff(H__field_(x, y, z, t), t))

(4.3)

Well that didn't work, so try to seperate the Curl of H so that we can substitute for D.``

collect(%, Curl(H__field_(x, y, z, t)))

Physics:-Vectors:-Curl(Physics:-Vectors:-Curl(E__field_(x, y, z, t))) = -mu*Physics:-Vectors:-Curl(diff(H__field_(x, y, z, t), t))

(4.4)

simplify(%)

Physics:-Vectors:-Curl(Physics:-Vectors:-Curl(E__field_(x, y, z, t))) = -mu*Physics:-Vectors:-Curl(diff(H__field_(x, y, z, t), t))

(4.5)

collect(%, Curl(H__field_(x, y, z, t)))

Physics:-Vectors:-Curl(Physics:-Vectors:-Curl(E__field_(x, y, z, t))) = -mu*Physics:-Vectors:-Curl(diff(H__field_(x, y, z, t), t))

(4.6)

``

``

``

SortProducts(%, [H__field_(x, y, z, t)], totheleft)

SortProducts(Physics:-Vectors:-Curl(Physics:-Vectors:-Curl(E__field_(x, y, z, t))) = -mu*Physics:-Vectors:-Curl(diff(H__field_(x, y, z, t), t)), [H__field_(x, y, z, t)], totheleft)

(4.7)

``

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``

isolate(%, H__field_(x, y, z, t))

SortProducts(Physics:-Vectors:-Curl(Physics:-Vectors:-Curl(E__field_(x, y, z, t))) = -mu*Physics:-Vectors:-Curl(diff(H__field_(x, y, z, t), t)), [H__field_(x, y, z, t)], totheleft) = 0

(4.8)

``

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NULL

NULL

> 

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I need to be able to rearrange equation 4.3 so that I can substitute for Curl of H using Maxwell_2.  Any suggestions would be gratefully received!
 

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