Question

B-Waves. Starting with Maxwell’s equations, derive the 3-D wave equation for magnetic fields.

Gauss's law for electric fields Gauss's law for magnetic fields: Faraday's law: (11-31a) (11-31b) (11-31c) OE Ampere's law (11-31d)

Answer 1

$\nabla \times B = \mu_0 \varepsilon_0 \frac{\partial \vec{E}}{\partial t}$

Taking curl on both sides, we get

$\nabla \times (\nabla \times B) = \mu_0 \varepsilon_0 \frac{\partial (\nabla \times \vec{E})}{\partial t}$

$\nabla (\nabla \cdot B) - \nabla ^2 B =$    $\mu_0 \varepsilon_0 \frac{\partial (\nabla \times \vec{E})}{\partial t}$

From equation (11.31-b), $\nabla \cdot B = 0$ and from equation (11.31-c), $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$

So,

$\nabla^2 \vec{B} = \mu_0 \varepsilon_0 \frac{\partial^2 \vec{B}}{\partial t^2 }$