Show Maxwell's equations in free space satisfy the wave equation
Maxwell's equations in free space:
∇ . E = 0
∇ x E = -∂B/∂t
∇ . B = 0
∇ x B = (1/c2)(∂E/∂t)
The wave equation:
∇2U = (1/c2)(∂2U/∂t2)
If we take the curl of ∇ x E, we get ∇ x(∇ x E) = -(∂/∂t)∇ x 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 -∇2E = -(∂/∂t)∇ x B
As ∇ x B = (1/c2)(∂E/∂t), we have -∇2E = -(∂/∂t)(1/c2)(∂E/∂t)
Thus, ∇2E = (1/c2)(∂2E/∂t2) which shows that Maxwell's equations satisfy the wave equation. A similar process can be applied to B
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