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 and the maximum electric field from the given magnetic field equation of an electromagnetic wave.

### Step-by-Step Solution:

1. **Identify the Magnetic Field Equation**:
The magnetic field is given as:
\[ B = 200 \, \mu T \, \sin(4 \times 10^{-5} \, s^{-1} (t - \frac{x}{c})) \]
Here, \( \mu T \) denotes microtesla, which is \( 10^{-6} \, T \).

2. **Extract the Maximum Magnetic Field**:
The standard form of the magnetic field in an electromagnetic wave is:
\[ B = B_0 \sin(kx - \omega t) \]
By comparing the given equation with the standard form, we can identify:
\[ B_0 = 200 \, \mu T = 200 \times 10^{-6} \, T \]
Thus, the maximum magnetic field \( B_{max} \) is:
\[ B_{max} = 200 \, \mu T = 200 \times 10^{-6} \, T \]

3. **Calculate the Maximum Electric Field**:
The relationship between the maximum electric field \( E_0 \) and the maximum magnetic field \( B_0 \) in an electromagnetic wave is given by:
\[ E_0 = c \cdot B_0 \]
where \( c \) is the speed of light in vacuum, approximately \( 3 \times 10^8 \, m/s \).

4. **Substitute the Values**:
Now substituting the values:
\[ E_0 = (3 \times 10^8 \, m/s) \cdot (200 \times 10^{-6} \, T) \]
\[ E_0 = 3 \times 200 \times 10^2 \, V/m \]
\[ E_0 = 600 \times 10^2 \, V/m = 6 \times 10^4 \, V/m \]

5. **Final Answers**:
- The maximum magnetic field \( B_{max} = 200 \, \mu T \) or \( 200 \times 10^{-6} \, T \).
- The maximum electric field \( E_{max} = 6 \times 10^4 \, V/m \).

### Summary of Results:
- Maximum Magnetic Field \( B_{max} = 200 \, \mu T \)
- Maximum Electric Field \( E_{max} = 6 \times 10^4 \, V/m \)