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Show Maxwell's equations in free space satisfy the wave equation

Maxwell's equations in free space:

∇ . E = 0

= -B/t

∇ . B = 0

∇ B = (1/c2)(∂E/t)

The wave equation: 

2(1/c2)(2U/t2)

If we take the curl of ∇ E, we get ∇ x(∇ E) = -(/t)∇ B

Using the vector formula a×(b×c) = b(a· c)−c(a·b), we can expand the left hand side to: ∇(∇ . E) - E(∇.∇)

Since ∇.E = 0, this becomes -2-(/t)∇ B

As ∇ B = (1/c2)(∂E/t), we have -2-(/t)(1/c2)(∂E/t)

Thus, 2(1/c2)(2E/t2) which shows that Maxwell's equations satisfy the wave equation. A similar process can be applied to B

Dojcin D. GCSE Physics tutor, A Level Chemistry tutor, GCSE Chemistry...

2 years ago

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