Question 705058

1. Identify the Given Parameters

• Power radiated isotropically, $P = 50 \text{ kW} = 50 \times 10^3 \text{ W}$
• Distance from the antenna, $R = 100 \text{ km} = 100 \times 10^3 \text{ m} = 10^5 \text{ m}$
• Speed of light, $c = 3 \times 10^8 \text{ m/s}$
• Permeability of free space, $\mu_0 = 4\pi \times 10^{-7} \text{ T}\cdot\text{m/A}$

2. Calculate the Intensity ($I$) of the Wave The problem states the antenna radiates "isotropically" (equally in all directions). Furthermore, the ground is a "perfect absorber." Because the ground does not reflect any waves, there is no reflected energy to interfere with or add to the direct wave. Therefore, the intensity at distance $R$ is identical to what it would be in free space. The energy is spread over a full spherical area $4\pi R^2$. $I = \frac{P}{4\pi R^2}$

$I = \frac{50 \times 10^3}{4\pi (10^5)^2}$

$I = \frac{50 \times 10^3}{4\pi \times 10^{10}} = \frac{50}{4\pi} \times 10^{-7} \text{ W/m}^2$

3. Relate Intensity to Peak Magnetic Field ($B_0$) The average intensity of a sinusoidal electromagnetic wave is related to its peak magnetic field by the standard formula: $I = \frac{c B_0^2}{2\mu_0}$

Rearranging this formula to solve for the peak magnetic field $B_0$: $B_0^2 = \frac{2\mu_0 I}{c}$

4. Execute the Calculation Substitute the expression for $I$ into the equation for $B_0^2$:

$B_0^2 = \frac{2\mu_0}{c} \left( \frac{P}{4\pi R^2} \right)$

$B_0^2 = \frac{2 (4\pi \times 10^{-7})}{3 \times 10^8} \left( \frac{50 \times 10^3}{4\pi \times 10^{10}} \right)$

Cancel $4\pi$ from the numerator and denominator:

$B_0^2 = \frac{2 \times 10^{-7} \times 50 \times 10^3}{3 \times 10^8 \times 10^{10}}$

$B_0^2 = \frac{100 \times 10^{-4}}{3 \times 10^{18}}$

$B_0^2 = \frac{10^{-2}}{3 \times 10^{18}}$

$B_0^2 = \frac{1}{3} \times 10^{-20}$

Now, take the square root to find $B_0$:

$B_0 = \sqrt{\frac{1}{3} \times 10^{-20}}$

$B_0 = \frac{1}{\sqrt{3}} \times 10^{-10}$

$B_0 \approx 0.577 \times 10^{-10} \text{ T}$

$B_0 \approx 5.77 \times 10^{-11} \text{ T}$

5. Conclusion The calculated peak magnetic field is approximately $5.77 \times 10^{-11} \text{ T}$. Looking at the given options, this value is closest to $5.5 \times 10^{-11}$.

Correct Option: 2