The magnetic component of a polarised wave of light is `[B_(x)=(4.0xx10^(-6)T)sin(1.57xx10^(7)m^(-1))y+omegat]`. The intensity of light is

Options

A. `1.9kW//m^(2)`
B. `3.8kW//m^(2)`
C. `5.7kW//m^(2)`
D. `7.6kW//m^(2)`

Explanation

To find the intensity of the light wave given the magnetic component of the polarized wave, we can follow these steps:

### Step 1: Identify the given magnetic field component
The magnetic field component of the polarized wave is given as:
\[ B_x = (4.0 \times 10^{-6} \, \text{T}) \sin(1.57 \times 10^{7} \, \text{m}^{-1} y + \omega t) \]

From this equation, we can identify the amplitude of the magnetic field:
\[ B_0 = 4.0 \times 10^{-6} \, \text{T} \]

### Step 2: Use the formula for intensity in terms of the magnetic field
The intensity \( I \) of an electromagnetic wave can be expressed in terms of the amplitude of the magnetic field \( B_0 \) using the formula:
\[ I = \frac{1}{2} \frac{B_0^2}{\mu_0} c \]
where:
- \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \)
- \( c \) is the speed of light, approximately \( 3 \times 10^8 \, \text{m/s} \)

### Step 3: Substitute the values into the intensity formula
Substituting the values into the formula:
1. \( B_0 = 4.0 \times 10^{-6} \, \text{T} \)
2. \( \mu_0 = 4\pi \times 10^{-7} \, \text{T m/A} \)
3. \( c = 3 \times 10^8 \, \text{m/s} \)

Thus, we have:
\[ I = \frac{1}{2} \frac{(4.0 \times 10^{-6})^2}{4\pi \times 10^{-7}} \times (3 \times 10^8) \]

### Step 4: Calculate \( B_0^2 \)
Calculating \( B_0^2 \):
\[ B_0^2 = (4.0 \times 10^{-6})^2 = 16.0 \times 10^{-12} \, \text{T}^2 \]

### Step 5: Substitute \( B_0^2 \) into the intensity formula
Now substituting \( B_0^2 \) into the intensity formula:
\[ I = \frac{1}{2} \frac{16.0 \times 10^{-12}}{4\pi \times 10^{-7}} \times (3 \times 10^8) \]

### Step 6: Calculate the intensity
Calculating the denominator:
\[ 4\pi \approx 12.566 \]
Thus:
\[ 4\pi \times 10^{-7} \approx 12.566 \times 10^{-7} \]

Now substituting:
\[ I = \frac{1}{2} \frac{16.0 \times 10^{-12}}{12.566 \times 10^{-7}} \times (3 \times 10^8) \]

Calculating:
\[ \frac{16.0 \times 10^{-12}}{12.566 \times 10^{-7}} \approx 1.273 \times 10^{-5} \]
Then:
\[ I \approx \frac{1}{2} \times 1.273 \times 10^{-5} \times 3 \times 10^8 \]
\[ I \approx 1.909 \times 10^{3} \, \text{W/m}^2 \approx 1900 \, \text{W/m}^2 \]

### Final Answer
The intensity of the light is approximately:
\[ I \approx 1900 \, \text{W/m}^2 \]