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**1. Homework Statement**

Consider an electromagnetic field in an empty space in the region ##0 \leq z \leq a## with the following non-zero components:

$$E_x = -B_0\frac{\omega a}{\pi}\sin\left(\frac{\pi z}{a}\right)\sin\left( ky-\omega t\right)\\

B_z = B_0\frac{ka}{\pi}\sin\left(\frac{\pi z}{a}\right)\sin\left( ky-\omega t\right)\\

B_y= B_0\cos\left(\frac{\pi z}{a}\right)\cos\left( ky-\omega t\right)

$$

Determine the condition for which this field satisfies Maxwell's equations. Assume that the fields are zero for z<0 and that there is a perfectly conducting plate in the z=0 plane and determine the surface charge density and surface current density on the plate.

**2. Homework Equations**

Maxwell's equations:

$$I. \quad \nabla \cdot \mathbf{E}=\frac{\rho}{\epsilon_0}\\

II. \quad \nabla \cdot \mathbf{B}=0\\

III. \quad \nabla \times \mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}\\

IV. \quad \nabla \times \mathbf{B}=\mu_0 \mathbf{J}+\mu_0\epsilon_0\frac{\partial \mathbf{E}}{\partial t}

$$

**3. The Attempt at a Solution**

It's the second part I have problems with. If you use MI (Maxwell I) you get ##\mathbf{E}=0 \Rightarrow \rho=0##. That's all fine and dandy, but if you use MIV you get

$$\nabla \times \mathbf{B}=\mu_0 \mathbf{J} = \left(\frac{k^2a}{\pi}+\frac{\pi}{a}\right)B_0\cos(ky-\omega t) \hat{\mathbf x}\\

\left(\frac{\partial \mathbf{E}}{\partial t}=0 \quad \text{at} \quad z=0\right).$$

So we've got a current density but no charge density and no electric field at ##z=0## even though I think there should be some, especially since we have a changing B-field which should induce an E-field. Something is obviously wrong with my thought process.