how does
E ̅(z,t)=|〖E_0〗_x a ̂_x |  cos⁡(ωt+ϕ_0 )
B ̅(z,t)=|〖B_0〗_y a ̂_y |  cos⁡(ωt+ϕ_0 )
equate to 
B(z,t)=B_0 a ̂_y cos(kz−ωt)

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.

how does E̅(z,t)=|〖E_0〗_x â_x| cos(ωt+ϕ_0 ) B̅(z,t)=|〖B_0〗_y â_y| cos(ωt+ϕ_0 ) equate to B(z,t)=B_0 â_y cos(kz−ωt)

Explore how the electric and magnetic fields in an electromagnetic wave, described by E̅(z,t) = E_0 â_x cos(ωt + ϕ_0) and B̅(z,t) = B_0 â_y cos(ωt + ϕ_0), equate to the traveling wave form B(z,t) = B_0 â_y cos(kz - ωt). This guide covers key concepts in wave propagation and the plane wave equation, suitable for university-level physics students.

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