An airplane flying at a distance of 10 km from a radio transmitter receives a signal of intensi

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An airplane flying at a distance of 10 km from a radio...

An airplane flying at a distance of $10 \mathrm{~km}$ from a radio transmitter receives a signal of intensity $10 \mu \mathrm{W} / \mathrm{m}^{2}$. What is the amplitude of the (a) electric and (b) magnetic component of the signal at the airplane? (c) If the transmitter radiates uniformly over a hemisphere, what is the transmission power?

Key Concepts

Electromagnetic Waves

Electromagnetic waves are oscillations of electric and magnetic fields that propagate through space at the speed of light. They are governed by Maxwell's equations and can carry energy, momentum, and information over long distances. The properties of these waves, such as wavelength, frequency, and amplitude, play a crucial role in the behavior and propagation of the wave.

Intensity of Electromagnetic Waves

The intensity of an electromagnetic wave is defined as the average power transmitted per unit area, and it is proportional to the square of the amplitude of the electric field. This concept is important because it relates the measurable energy flux at a point to the properties of the wave's fields, enabling calculations involving energy transfer in various applications.

Relationship Between Electric and Magnetic Fields

In free space, the electric and magnetic fields in an electromagnetic wave are intrinsically linked such that their magnitudes are related by the speed of light (c). Specifically, the magnetic field amplitude can be obtained by dividing the electric field amplitude by c. This relationship is fundamental in understanding the dynamics of electromagnetic wave propagation.

Poynting Vector

The Poynting vector represents the directional energy flux (the rate of energy transfer per unit area) of an electromagnetic field. It is defined as the cross product of the electric and magnetic fields, and its time-averaged magnitude is directly related to the intensity of the wave. This concept is essential for relating the wave’s field amplitudes to the power density it carries.

Radiation Distribution and Transmission Power

The transmission power of a radiating source can be determined by considering how the energy spreads out over a surface. When a transmitter radiates uniformly over a hemisphere, the total transmitted power is calculated by multiplying the intensity by the area of the hemispherical surface over which the energy is distributed. This concept is crucial for understanding and designing systems that depend on uniform energy distribution from a source.

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