The magnetic field in a plane electromagnetic wave is given by `B = 200 (muT) sin 4 xx 10'^(-5)s^(1) (t-x//c)`. Find the maximum magnetic and electric fields.

Answer

To solve the problem, we need to find the maximum magnetic field (B₀) and the corresponding maximum electric field (E₀) in the given electromagnetic wave.

### Step-by-step Solution:

1. **Identify the Maximum Magnetic Field (B₀)**:
The magnetic field is given by the equation:
\[ B = 200 \, \mu T \, \sin(4 \times 10^{-5} \, s^{-1} \, (t - \frac{x}{c})) \]
Here, \(B₀\) (the maximum magnetic field) is the coefficient of the sine function. Thus,
\[ B₀ = 200 \, \mu T = 200 \times 10^{-6} \, T \]

2. **Convert Microtesla to Tesla**:
We have already converted microtesla to tesla:
\[ B₀ = 200 \times 10^{-6} \, T \]

3. **Use the Relationship Between B₀ and E₀**:
In electromagnetic waves, the relationship between the maximum electric field (E₀) and the maximum magnetic field (B₀) is given by:
\[ E₀ = B₀ \cdot c \]
where \(c\) is the speed of light, approximately \(3 \times 10^8 \, m/s\).

4. **Calculate E₀**:
Substitute the values of \(B₀\) and \(c\) into the equation:
\[ E₀ = (200 \times 10^{-6} \, T) \cdot (3 \times 10^8 \, m/s) \]

5. **Perform the Calculation**:
\[ E₀ = 200 \times 3 \times 10^{2} \, V/m = 600 \times 10^{2} \, V/m = 6 \times 10^{4} \, V/m \]

### Final Answers:
- Maximum Magnetic Field (B₀): \(200 \, \mu T\) or \(200 \times 10^{-6} \, T\)
- Maximum Electric Field (E₀): \(6 \times 10^{4} \, V/m\)

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