Question

Using Maxwell’s equations in free space, derive the wave equation for the

electric field vector. Obtain the conditions under which the following time varying electric and magnetic fields satisfy Maxwell’s equations in vacuum with no source charges or currents:

E = kˆ E0 sin ( y − vt)

B = ˆi B0 sin ( y − vt)

where E0 and B0 are constant.

27 Jan 2018
Answer :
Word Count : 757
We start with Maxwell’s equations in free space (vacuum) where there are no free charges or currents: \[ \nabla \cdot \mathbf{E} = 0 \] \[ \nabla \cdot \mathbf{B} = 0 \] \[ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} \] \[ \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \] where \(\epsilon_0\) is the permittivity of free space, and \(\mu_0\) is the permeability of free space. --- ### Step 1: Deriving the Wave Equation for \( \mathbf{E} \) Taking the curl of Faraday’s law: \[ \nabla \times (\nabla \times \mathbf{E}) = -\nabla \times \frac{\partial \mathbf{B}}{\partial t} \] Using the vector identity: \[ \nabla \times (\nabla \times \mathbf{E}) = \nabla (\nabla \cdot \mathbf{E}) - \nabla^2 \mathbf{E} \] Since \( \nabla \cdot \mathbf{E} = 0 \), this simplifies to: \[ -\nabla^2 \mathbf{E} = -\frac{\partial}{\partial t} (\nabla \times \mathbf{B}) \] Using Ampère’s law: \[ \nabla \times \mathbf{B} = \mu_0 \epsilon_0 \frac{\partial \mathbf{E}}{\partial t} \] we substitute: \[ \nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{\partial^2 \mathbf{E}}{\partial t^2} \] This is the wave equation for the electric field: \[
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