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 can follow these steps:

### Step 1: Identify the wave number (k)
The magnetic field is given by:
\[ B = 3.01 \times 10^{-7} \sin(6.28 \times 10^{2} x + 2.2 \times 10^{10} t) \]

In the standard form of a wave equation:
\[ B = B_0 \sin(kx - \omega t) \]
we can see that the coefficient of \( x \) in the sine function is \( k \).

From the given equation, we have:
\[ k = 6.28 \times 10^{2} \, \text{cm}^{-1} \]

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

### Step 3: Solve for wavelength
We can rearrange the equation to solve for \( \lambda \):
\[ \lambda = \frac{2\pi}{k} \]

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

### Step 4: Calculate \( \lambda \)
Now, we can calculate \( \lambda \):
\[ \lambda = \frac{2 \times 3.14}{6.28 \times 10^{2}} \]

This simplifies to:
\[ \lambda = \frac{6.28}{6.28 \times 10^{2}} = \frac{1}{10^{2}} \]

### Step 5: Convert to appropriate units
Since \( \lambda \) is in meters (as \( k \) is in cm\(^{-1}\)), we convert:
\[ \lambda = 0.01 \, \text{m} = 1 \, \text{cm} \]

### Final Answer
The wavelength of the given wave is:
\[ \lambda = 1 \, \text{cm} \]

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