The electric field of an EM wave is given by \(E_{z}=\) $E_{\mathrm{m}} \sin (k y-\omega t+\pi / 6), E_{x}=0,\( and \)E_{z}=0 .$ (a) In what direction is this wave traveling? (b) Write expressions for the components of the magnetic field of this wave.

The given electric field of the EM wave is $$ E_{z} = E_{\mathrm{m}} \sin (ky - \omega t + \pi / 6), \quad E_{x} = 0, \quad E_{y} = 0. $$ The equation for \(E_z\) indicates that we have a wave traveling in the \(\boldsymbol{y}\)-direction. The general form of the wave is \(E (\boldsymbol{r}, t) = E_m\sin (k\boldsymbol{(r \cdot y)} - \omega t + \phi)\) (assuming wave propagation along \(\boldsymbol{y}\)-axis). Therefore, the direction of the wave is along \(\boldsymbol{y}\).

For this part, we'll use Maxwell's equations, which relate the electric and magnetic fields. Looking at the Faraday's Law and the Ampere's Law with Maxwell's addition, we get: $$ \nabla \times \boldsymbol{E} = - \frac{\partial \boldsymbol{B}}{\partial t}, \quad \nabla \times \boldsymbol{B} = \mu_0 \epsilon_0 \frac{\partial \boldsymbol{E}}{\partial t}. $$ Let's first calculate the curl for the electric field, \(\nabla \times \boldsymbol{E}\). Since \(E_x = E_y = 0\), the curl can be simplified as: $$ \nabla \times \boldsymbol{E} = \left(\frac{\partial E_z}{\partial y} - 0\right)\boldsymbol{x} - \frac{\partial E_z}{\partial x}\boldsymbol{y} + 0\boldsymbol{z}. $$ Computing the partial derivative of \(E_z\), we get: $$ \frac{\partial E_z}{\partial y} = kE_m\cos(ky-\omega t+\pi/6). $$ Now, we can find the curl: $$ \nabla \times \boldsymbol{E} = kE_m\cos(ky - \omega t + \pi/6)\boldsymbol{x}. $$ Using Faraday's Law, we can determine \(\frac{\partial \boldsymbol{B}}{\partial t}\): $$ \frac{\partial \boldsymbol{B}}{\partial t} = -kE_m\cos(ky - \omega t + \pi/6)\boldsymbol{x}. $$ From this equation, we can see that the only non-zero magnetic component is \(B_x\). So let's calculate this component by integrating with respect to time: $$ B_x(y,t) = \int -kE_m\cos(ky - \omega t + \pi/6) dt. $$ Therefore, the \(B_x\) component is: $$ B_x(y,t) = \frac{kE_m}{\omega}\sin(ky - \omega t + \pi/6) + C(y) $$ where \(C(y)\) is an integration constant, which can be assumed to be zero for most situations. The final expressions for the magnetic fields are: $$ B_x(y,t) = \frac{kE_m}{\omega}\sin(ky - \omega t + \pi/6), \quad B_y = 0, \quad B_z = 0. $$