The intensity of `TV` broad cast station of `E=800sin(10^(9)t-kx) V//M` is…….and the wave length in meter is…….

Options

A. `850 wm^(-2)`, `0.6 pi`
B. `425 wm^(-2)`, `0.6 pi`
C. `850 wm^(-2)`, `0.3 pi`
D. `425 wm^(-2)`, `0.3 pi`

Explanation

To solve the problem, we need to determine the intensity and wavelength of the electromagnetic wave given by the equation \( E = 800 \sin(10^9 t - kx) \) V/m.

### Step 1: Identify the parameters from the wave equation
The given wave equation is in the form \( E = E_0 \sin(\omega t - kx) \), where:
- \( E_0 = 800 \) V/m (the amplitude of the electric field)
- \( \omega = 10^9 \) rad/s (the angular frequency)

### Step 2: Calculate the wavelength (\( \lambda \))
The relationship between the angular frequency (\( \omega \)) and the wavelength (\( \lambda \)) is given by:
\[ \lambda = \frac{c}{\nu} \]
where \( \nu = \frac{\omega}{2\pi} \) is the frequency and \( c \) is the speed of light (\( c \approx 3 \times 10^8 \) m/s).

First, calculate the frequency (\( \nu \)):
\[ \nu = \frac{\omega}{2\pi} = \frac{10^9}{2\pi} \]

Now, substitute \( \nu \) into the wavelength formula:
\[ \lambda = \frac{c}{\nu} = \frac{3 \times 10^8}{\frac{10^9}{2\pi}} = \frac{3 \times 10^8 \cdot 2\pi}{10^9} \]

This simplifies to:
\[ \lambda = \frac{6\pi}{10} = 0.6\pi \text{ meters} \]

### Step 3: Calculate the intensity (\( I \))
The intensity of an electromagnetic wave is given by the formula:
\[ I = \frac{E_0^2}{2 \mu_0 c} \]
where \( \mu_0 \) (the permeability of free space) is approximately \( 4\pi \times 10^{-7} \, \text{T m/A} \).

Substituting the values:
\[ I = \frac{800^2}{2 \cdot (4\pi \times 10^{-7}) \cdot (3 \times 10^8)} \]

Calculating \( 800^2 \):
\[ 800^2 = 640000 \]

Now substitute this into the intensity formula:
\[ I = \frac{640000}{2 \cdot (4\pi \times 10^{-7}) \cdot (3 \times 10^8)} \]

Calculating the denominator:
\[ 2 \cdot (4\pi \times 10^{-7}) \cdot (3 \times 10^8) = 24\pi \times 10^{-7} \times 10^8 = 24\pi \times 10^{-1} = 2.4\pi \]

Thus, we have:
\[ I = \frac{640000}{2.4\pi} \]

Calculating the final intensity:
\[ I \approx \frac{640000}{7.54} \approx 84800 \text{ W/m}^2 \]

### Final Answers:
- The intensity of the electromagnetic wave is approximately \( 84800 \, \text{W/m}^2 \).
- The wavelength of the wave is \( 0.6\pi \, \text{meters} \).