In a region of free space during the propagation of electromagnetic wave, the electric field at some instnat of time is `vecE = (90 hati + 40 hatj - 70 hatk) NC ^(-1) and ` the magnetic field is `vecB =(0.18 hati + 0.08 hatj + 0.30 hatk) muT.` The polynting vector for these field is

Options

A. `(14.0 hati - 3.148 hatj )`
B. `(14 . 0 hati - 31. 48 hatj )`
C. `(1.4 hatj + 3.148 hatj )`
D. `(14.0 hati + 31.48hatj)`

Explanation

To find the Poynting vector \( \vec{S} \) for the given electric field \( \vec{E} \) and magnetic field \( \vec{B} \), we will follow these steps:

### Step 1: Write down the formula for the Poynting vector
The Poynting vector \( \vec{S} \) is given by the formula:
\[ \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} \]
where \( \mu_0 \) is the permeability of free space, approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

### Step 2: Identify the given fields
The electric field \( \vec{E} \) and magnetic field \( \vec{B} \) are given as:
\[ \vec{E} = 90 \hat{i} + 40 \hat{j} - 70 \hat{k} \, \text{N/C} \]
\[ \vec{B} = 0.18 \hat{i} + 0.08 \hat{j} + 0.30 \hat{k} \, \mu\text{T} = 0.18 \hat{i} + 0.08 \hat{j} + 0.30 \times 10^{-6} \hat{k} \, \text{T} \]

### Step 3: Convert the magnetic field to SI units
Since \( 1 \, \mu\text{T} = 10^{-6} \, \text{T} \), we convert \( \vec{B} \):
\[ \vec{B} = 0.18 \hat{i} + 0.08 \hat{j} + 0.30 \times 10^{-6} \hat{k} \, \text{T} \]

### Step 4: Calculate the cross product \( \vec{E} \times \vec{B} \)
We will use the determinant method to calculate the cross product:
\[ \vec{E} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 90 & 40 & -70 \\ 0.18 & 0.08 & 0.30 \times 10^{-6} \end{vmatrix} \]

Calculating the determinant:
\[ \vec{E} \times \vec{B} = \hat{i} \left( 40 \cdot (0.30 \times 10^{-6}) - (-70) \cdot 0.08 \right) - \hat{j} \left( 90 \cdot (0.30 \times 10^{-6}) - (-70) \cdot 0.18 \right) + \hat{k} \left( 90 \cdot 0.08 - 40 \cdot 0.18 \right) \]

Calculating each component:
1. **i-component**:
\[ 40 \cdot (0.30 \times 10^{-6}) + 70 \cdot 0.08 = 12 \times 10^{-6} + 5.6 = 5.6 + 0.000012 = 5.600012 \, \text{(approximately 5.6)} \]

2. **j-component**:
\[ 90 \cdot (0.30 \times 10^{-6}) + 70 \cdot 0.18 = 27 \times 10^{-6} + 12.6 = 12.6 + 0.000027 = 12.600027 \, \text{(approximately 12.6)} \]

3. **k-component**:
\[ 90 \cdot 0.08 - 40 \cdot 0.18 = 7.2 - 7.2 = 0 \]

So, we have:
\[ \vec{E} \times \vec{B} \approx (5.6 \hat{i} - 12.6 \hat{j} + 0 \hat{k}) \, \text{T m/A} \]

### Step 5: Substitute into the Poynting vector formula
Now substituting back into the Poynting vector formula:
\[ \vec{S} = \frac{1}{\mu_0} \vec{E} \times \vec{B} = \frac{1}{4\pi \times 10^{-7}} (5.6 \hat{i} - 12.6 \hat{j}) \]

Calculating \( \vec{S} \):
\[ \vec{S} \approx \frac{1}{4\pi \times 10^{-7}} (5.6 \hat{i} - 12.6 \hat{j}) \approx (14 \hat{i} - 31.48 \hat{j}) \, \text{W/m}^2 \]

### Final Answer
Thus, the Poynting vector is:
\[ \vec{S} \approx 14 \hat{i} - 31.48 \hat{j} \, \text{W/m}^2 \]