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The electric field amplitude in an electromagnetic wave is a measure of the maximum strength of the electric field in that wave. Understanding how to find this amplitude is crucial when dealing with wave intensity, such as the signal strength received by an airplane from a radio transmitter.
To calculate the amplitude of the electric field (\(E_m\)), it's important to relate it to wave intensity (\(I\)). The formula we use is:

• \[ I = \frac{c\epsilon_0}{2}E_m^2 \]

Here, \(c\) is the speed of light (\(3 \times 10^8 \text{ m/s}\)), and \(\epsilon_0\) represents the permittivity of free space (\(8.85 \times 10^{-12} \text{ F/m}\)).
In the given problem, the intensity is provided as \(10 \mu \text{W/m}^2\). We rearrange the equation to solve for \(E_m\):

• \[ E_m = \sqrt{\frac{2I}{c\epsilon_0}} \]

Plugging in the known values, we find the electric field amplitude to be \(1.94 \text{ V/m}\). This calculation demonstrates that by knowing the wave's intensity, one can determine the electric field's strength at any point in space.

Magnetic field amplitude is directly related to electric field amplitude when dealing with electromagnetic waves, such as radio waves. It represents the maximum strength of the magnetic field in the wave.
To find the magnetic field amplitude (\(B_m\)), we utilize the relationship it holds with the electric field amplitude. Given they are components of the same wave, they're connected by the speed of light (\(c\)). The key formula here is:

• \[ B_m = \frac{E_m}{c} \]

This relationship is straightforward because electromagnetic waves propagate with their electric and magnetic fields perpendicular to each other. Given the previously calculated \(E_m\) as \(1.94 \text{ V/m}\), substituting into the formula, we find:

• \[ B_m = \frac{1.94}{3 \times 10^8} = 6.47 \times 10^{-9} \text{ T} \]

This tells us the strength of the magnetic field component in the radio wave, illustrating how powerful a magnetic influence will accompany the electric component.

Transmission power is how we measure the total energy output of a transmitter over a given area, like a hemisphere surrounding a radio transmitter.
To find the transmission power (\(P\)), we relate it to intensity (\(I\)) and area (\(A\)) through the equation:

• \[ I = \frac{P}{A} \]

Here, the transmitter radiates uniformly over a hemisphere. This area is given by \(2\pi r^2\), with \(r\) being the distance from the transmitter \(10 \text{ km}\) (or \(10 \times 10^3 \text{ m}\)).
Substituting in the given intensity and solving for power, we use:

• \[ P = I \times 2\pi r^2 \]

After substitution:

• \[ P = 10 \times 10^{-6} \times 2\pi \times (10 \times 10^3)^2 = 6.28 \text{ kW} \]

This result shows the total power distributed across the hemisphere, indicating how much energy is being transmitted through space as a radio wave.