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

Answer

To solve the problem of finding the wavelength of the electromagnetic wave given by the magnetic field \( B_y = 2 \times 10^{-7} \sin(\pi \times 10^3 x + 3\pi \times 10^{11} t) \, T \), we can follow these steps:

### Step 1: Identify the wave equation format
The magnetic field is given in the form:
\[ B_y = B_0 \sin(kx + \omega t) \]
where \( B_0 \) is the maximum magnetic field, \( k \) is the wave number, and \( \omega \) is the angular frequency.

### Step 2: Extract the wave number \( k \)
From the given equation, we can identify the coefficient of \( x \) in the sine function:
\[ k = \pi \times 10^3 \, \text{m}^{-1} \]

### Step 3: Relate wave number \( k \) to wavelength \( \lambda \)
The wave number \( k \) is related to the wavelength \( \lambda \) by the formula:
\[ k = \frac{2\pi}{\lambda} \]
We can rearrange this to find \( \lambda \):
\[ \lambda = \frac{2\pi}{k} \]

### Step 4: Substitute the value of \( k \) into the wavelength formula
Substituting the value of \( k \):
\[ \lambda = \frac{2\pi}{\pi \times 10^3} \]

### Step 5: Simplify the expression
The \( \pi \) in the numerator and denominator cancels out:
\[ \lambda = \frac{2}{10^3} = 2 \times 10^{-3} \, \text{m} \]

### Step 6: Final answer
Thus, the wavelength \( \lambda \) of the electromagnetic wave is:
\[ \lambda = 2 \times 10^{-3} \, \text{m} \, \text{or} \, 2 \, \text{mm} \]

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