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

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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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