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. a visible light
B. an infrared wave
C. a microwave
D. a radio wave

Explanation

To determine the type of electromagnetic wave represented by the given magnetic field equation, we will follow these steps:

### Step 1: Identify the given magnetic field equation
The magnetic field in the plane electromagnetic wave is given by:
\[ B_y = 2 \times 10^{-7} \sin(0.5 \times 10^{3} x + 1.5 \times 10^{11} t) \]

### Step 2: Extract the wave vector (k) and angular frequency (ω)
From the general form of the wave equation \( B_y = B_0 \sin(kx + \omega t) \), we can identify:
- Amplitude \( B_0 = 2 \times 10^{-7} \)
- Wave vector \( k = 0.5 \times 10^{3} \) (in rad/m)
- Angular frequency \( \omega = 1.5 \times 10^{11} \) (in rad/s)

### Step 3: Calculate the wavelength (λ)
The relationship between the wave vector \( k \) and the wavelength \( λ \) is given by:
\[ k = \frac{2\pi}{\lambda} \]
Rearranging gives:
\[ \lambda = \frac{2\pi}{k} \]

Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{0.5 \times 10^{3}} \]
\[ \lambda = \frac{2 \times 3.14}{0.5 \times 10^{3}} \]
\[ \lambda = \frac{6.28}{0.5 \times 10^{3}} \]
\[ \lambda = \frac{6.28}{500} \]
\[ \lambda = 0.01256 \text{ m} = 12.56 \text{ mm} \]

### Step 4: Determine the type of electromagnetic wave
Now that we have the wavelength \( \lambda = 12.56 \text{ mm} \), we can classify the electromagnetic wave based on its wavelength:
- Radio waves: \( > 1 \text{ m} \)
- Microwaves: \( 1 \text{ m} \) to \( 1 \text{ mm} \)
- Infrared: \( 1 \text{ mm} \) to \( 700 \text{ nm} \)
- Visible light: \( 700 \text{ nm} \) to \( 400 \text{ nm} \)
- Ultraviolet: \( 400 \text{ nm} \) to \( 10 \text{ nm} \)
- X-rays: \( 10 \text{ nm} \) to \( 0.01 \text{ nm} \)
- Gamma rays: \( < 0.01 \text{ nm} \)

Since \( 12.56 \text{ mm} \) falls within the microwave range, we conclude that the electromagnetic wave is a **microwave**.

### Final Answer
The electromagnetic wave is a **microwave**.

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