https://file.aiquickdraw.com/mathbot/file/1730318968718hx69ve8b.jpg

What is the wavelength of the wave?

Express your answer to two significant figures and include the appropriate units.

To find the wavelength \( \lambda \) of the electromagnetic wave, we can use the wave vector \( k \) given in the electric field equation. 

The general form for the electric field of a wave traveling in the \( z \)-direction is:
\[
\vec{E} = E_0 \sin(kz - \omega t)
\]
where:
- \( k \) is the wave number, and it’s related to the wavelength \( \lambda \) by the formula \( k = \frac{2\pi}{\lambda} \).

In the equation provided in the image:
\[
\vec{E} = \hat{i} (225 \, \text{V/m}) \sin \left( (0.077 \, \text{m}^{-1}) z - (2.3 \times 10^7 \, \text{rad/s}) t \right)
\]

From this, we can see that \( k = 0.077 \, \text{m}^{-1} \).

### Step-by-Step Solution

1. **Use the relationship between \( k \) and \( \lambda \):**
   \[
   \lambda = \frac{2\pi}{k}
   \]

2. **Substitute the given value of \( k \):**
   \[
   \lambda = \frac{2\pi}{0.077 \, \text{m}^{-1}}
   \]

3. **Calculate \( \lambda \):**
   \[
   \lambda \approx \frac{6.2832}{0.077} \approx 82 \, \text{m}
   \]

### Answer
The wavelength \( \lambda \) of the wave is approximately:
\[
\lambda = 82 \, \text{m}
\]

Would you like more details or have any other questions?

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Here are some related questions to further explore:

1. What is the frequency of this wave, given the angular frequency \( \omega \)?
2. How would you calculate the speed of the wave using \( \lambda \) and the frequency?
3. What is the magnetic field component of this wave?
4. How does the amplitude of the electric field relate to the intensity of the wave?
5. What effect would doubling the wave vector \( k \) have on the wavelength?

**Tip:** Remember, the wavelength is inversely proportional to the wave number \( k \); larger \( k \) means a shorter wavelength.

What is the wavelength of the wave? Express your answer to two significant figures and include the appropriate units.

Find the wavelength of an electromagnetic wave from its electric field equation. This problem involves using the wave vector and its relationship with wavelength, suitable for college-level physics students.

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