The magnetic field in a plane em wave is given by `B_y = 2 x 10 ^(-7) sin (pi x 10^3 x + 3 pi x 10^11 t )T Calculate the wavelength

Options

A. `pi *10^3`
B. `2 *10^-3 m`
C. `2 *10^3 m`
D. `pi *10^-3 m`

Answer

To calculate the wavelength from the given magnetic field of a plane electromagnetic (EM) wave, we can follow these steps:

### Step 1: Identify the given magnetic field equation
The magnetic field is given as:
\[ B_y = 2 \times 10^{-7} \sin\left( \pi \times 10^{3} x + 3 \pi \times 10^{11} t \right) \]

### Step 2: Compare with the general form of the magnetic field
The general form of the magnetic field in an EM wave can be expressed as:
\[ B = B_0 \sin(kx - \omega t) \]
where:
- \( B_0 \) is the amplitude,
- \( k \) is the wave number,
- \( \omega \) is the angular frequency.

### Step 3: Identify the wave number \( k \) and angular frequency \( \omega \)
From the given equation, we can identify:
- The term multiplying \( x \) in the sine function is \( \pi \times 10^{3} \), which gives us:
\[ k = \pi \times 10^{3} \, \text{m}^{-1} \]
- The term multiplying \( t \) is \( 3\pi \times 10^{11} \), which gives us:
\[ \omega = 3\pi \times 10^{11} \, \text{s}^{-1} \]

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

### Step 5: Solve for wavelength \( \lambda \)
Rearranging the equation for \( \lambda \):
\[ \lambda = \frac{2\pi}{k} \]

Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{\pi \times 10^{3}} \]

### Step 6: Simplify the expression
\[ \lambda = \frac{2}{10^{3}} \]
\[ \lambda = 2 \times 10^{-3} \, \text{m} \]

### Final Answer
The wavelength \( \lambda \) is:
\[ \lambda = 2 \times 10^{-3} \, \text{m} \, \text{or} \, 2 \, \text{mm} \]

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