The equations you're working with describe the electric field $\vec{E}(z,t)$ and magnetic field $\vec{B}(z,t)$ in a plane electromagnetic wave.

Let's analyze the equations one step at a time and figure out how the given $\vec{E}(z,t)$ and $\vec{B}(z,t)$ expressions lead to the simplified form for $\vec{B}(z,t)$.

1. Initial Forms:

You have the initial electric and magnetic fields:

$\vec{E}(z,t) = |E_{0x} \hat{a}_x| \cos(\omega t + \varphi_0)$ $\vec{B}(z,t) = |B_{0y} \hat{a}_y| \cos(\omega t + \varphi_0)$

These represent oscillations of the electric and magnetic fields, where:

• $E_{0x}$ and $B_{0y}$ are the amplitudes of the electric and magnetic fields, respectively.
• $\omega$ is the angular frequency of the wave.
• $\varphi_0$ is an initial phase.
• The fields are polarized along $\hat{a}_x$ and $\hat{a}_y$, respectively.

2. Simplified Magnetic Field Expression:

You want to show how the second equation for $\vec{B}(z,t)$:

$\vec{B}(z,t) = B_0 \hat{a}_y \cos(kz - \omega t)$

arises. This is a standard expression for a traveling plane wave propagating in the $z$-direction, where:

• $k$ is the wave number, and it is related to the wavelength $\lambda$ by $k = \frac{2\pi}{\lambda}$.
• $\omega t$ represents the time-dependent oscillation.
• $kz - \omega t$ shows that the wave is traveling in the positive $z$-direction.

3. Reconciling the Two Forms:

• In the initial expression for $\vec{B}(z,t)$, there is no explicit dependence on $z$, only $t$. This suggests that the initial form represents the field at a specific point in space, or it's a simplified snapshot.

• To convert to a traveling wave form, we need to introduce the spatial dependence on $z$. This is done by assuming that the wave propagates in the $z$-direction, leading to the phase factor $kz - \omega t$.

• The phase shift $\varphi_0$ in the original equation can be absorbed into the wave phase $kz - \omega t$ by choosing the appropriate value of $z$ or adjusting the initial time $t_0$. Thus, the overall phase can be modified to fit the traveling wave form $kz - \omega t$.

4. Conclusion:

The transformation from:

$\vec{B}(z,t) = |B_0 \hat{a}_y| \cos(\omega t + \varphi_0)$

to:

$\vec{B}(z,t) = B_0 \hat{a}_y \cos(kz - \omega t)$

is essentially the inclusion of the spatial dependence $kz$, which corresponds to the wave traveling along the $z$-axis. The initial phase $\varphi_0$ can be set to zero or incorporated into the wave's spatial dependence. The result is a typical expression for a plane electromagnetic wave.

Would you like more details, or do you have any specific questions on this?

Related Questions:

1. How does the wave number $k$ relate to the frequency $\omega$ in an electromagnetic wave?
2. What is the relationship between the electric and magnetic fields in a traveling electromagnetic wave?
3. How do we determine the direction of wave propagation from the phase factor $kz - \omega t$?
4. Why can we absorb the phase $\varphi_0$ into the term $kz - \omega t$?
5. How does polarization affect the electric and magnetic field vectors in an EM wave?

Tip:

In electromagnetic waves, the electric and magnetic fields are always perpendicular to each other and to the direction of wave propagation. This is a key feature of transverse waves.