The magnetic field in a plane electromagnetic wave is given by
`B_(y) = 2 xx 10^(-7) sin (0.5 xx 10^(3)x + 1.5 xx 10^(11) t)`.
This electromagnetic wave is

Options

A. Visible light
B. Infared
C. Microwave
D. Radiowaves

Answer

To determine the type of electromagnetic wave given the magnetic field equation \( B_y = 2 \times 10^{-7} \sin(0.5 \times 10^3 x + 1.5 \times 10^{11} t) \), we will follow these steps:

### Step 1: Identify the wave number \( k \)
The wave number \( k \) is the coefficient of \( x \) in the sine function. From the given equation, we can see that:
\[ k = 0.5 \times 10^3 \, \text{m}^{-1} \]

### Step 2: Relate wave number to wavelength \( \lambda \)
The relationship between the wave number \( k \) and the wavelength \( \lambda \) is given by the formula:
\[ k = \frac{2\pi}{\lambda} \]
We can rearrange this to find \( \lambda \):
\[ \lambda = \frac{2\pi}{k} \]

### Step 3: Calculate the wavelength \( \lambda \)
Substituting the value of \( k \) into the equation:
\[ \lambda = \frac{2\pi}{0.5 \times 10^3} \]
Calculating this gives:
\[ \lambda = \frac{2 \times 3.14}{0.5 \times 10^3} = \frac{6.28}{0.5 \times 10^3} = \frac{6.28}{500} = 0.01256 \, \text{m} = 12.56 \, \text{mm} \]

### Step 4: Determine the type of electromagnetic wave
Now that we have the wavelength \( \lambda \), we can classify the electromagnetic wave. The ranges for different types of electromagnetic waves are approximately:
- Radio Waves: \( \lambda > 1 \, \text{m} \)
- Microwaves: \( 1 \, \text{m} > \lambda > 1 \, \text{mm} \)
- Infrared: \( 1 \, \text{mm} > \lambda > 700 \, \text{nm} \)
- Visible Light: \( 700 \, \text{nm} > \lambda > 400 \, \text{nm} \)

Since \( \lambda = 12.56 \, \text{mm} \), it falls within the microwave range.

### Conclusion
Thus, the electromagnetic wave described by the given magnetic field equation is a **microwave**.

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