Question

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.

Short answer

Question: Determine the direction of the electromagnetic wave and find expressions for the components of the magnetic field. Answer: The direction of the electromagnetic wave is along the y-axis. The expressions for the components of the magnetic field are: \(B_x(y,t) = \frac{kE_m}{\omega}\sin(ky - \omega t + \pi/6), B_y = 0, B_z = 0\).

Step-by-step solution

Part (a) - Determine the direction the wave is traveling

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 \(\boldsymbol{y}\).

Part (b) - Find expressions for the components of the magnetic field

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: $$ B_x(y,t) = \frac{kE_m}{\omega}\sin(ky - \omega t + \pi/6), \quad B_y = 0, \quad B_z = 0. $$