The magnetic field in plane electromagnetic wave is given by `B= 3.01 xx 10^(-7) sin (6.28 xx 10^(2)x + 2.2 xx 10^(10) t)` I. [ where x in cm and t in second] The wavelength of the given wave is

Options

A. 1 cm
B. 628 cm
C. 1.129 cm
D. 3.14 cm

Answer

To find the wavelength of the given electromagnetic wave, we will follow these steps:

### Step-by-Step Solution:

1. **Identify the given magnetic field equation**:
The magnetic field is given by:
\[ B = 3.01 \times 10^{-7} \sin(6.28 \times 10^{2} x + 2.2 \times 10^{10} t) \]
Here, \( x \) is in centimeters and \( t \) is in seconds.

2. **Recognize the standard form of the wave equation**:
The standard form of the magnetic field in a plane electromagnetic wave is:
\[ B = B_0 \sin(kx - \omega t) \]
where \( k \) is the wave number and \( \omega \) is the angular frequency.

3. **Extract the wave number \( k \)**:
From the given equation, we can identify:
\[ k = 6.28 \times 10^{2} \, \text{(in cm}^{-1}\text{)} \]

4. **Relate wave number \( k \) to wavelength \( \lambda \)**:
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]

5. **Rearranging the formula to find \( \lambda \)**:
We can rearrange the equation to solve for \( \lambda \):
\[ \lambda = \frac{2\pi}{k} \]

6. **Substituting the value of \( k \)**:
Now substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{6.28 \times 10^{2}} \]

7. **Calculating \( \lambda \)**:
Using \( \pi \approx 3.14 \):
\[ \lambda = \frac{2 \times 3.14}{6.28 \times 10^{2}} = \frac{6.28}{6.28 \times 10^{2}} = \frac{1}{10^{2}} = 0.01 \, \text{m} \]

8. **Convert meters to centimeters**:
Since \( 1 \, \text{m} = 100 \, \text{cm} \):
\[ \lambda = 0.01 \, \text{m} = 1 \, \text{cm} \]

9. **Conclusion**:
Therefore, the wavelength \( \lambda \) of the given wave is:
\[ \lambda = 1 \, \text{cm} \]