Question

Isotropic electromagnetic waves expand uniformly outward in all directions in three dimensions. Electromagnetic waves from a small, isotropic source are not plane waves, which have constant maximum amplitudes. a) How does the maximum amplitude of the electric field of radiation from a small, isotropic source vary with distance from the source? b) Compare this with the electrostatic field of a point charge.

Short answer

Answer: The maximum amplitude of the electric field of radiation from an isotropic source varies with distance following an inverse square law, decreasing with the inverse square of the distance from the source. This behavior is similar to the electrostatic field of a point charge, which also follows an inverse square law, decreasing with the inverse square of the distance from the charge.

Step-by-step solution

Calculate the intensity of an electromagnetic wave

The intensity of an electromagnetic wave is defined as the power per unit area (usually given in Watts per square meter). The total power of an isotropic source is distributed equally over the surface of an expanding sphere centered at the source. The intensity, I, is given by: I = \frac{P}{A} where P is the power of the source and A is the surface area of the sphere.

Determine the relationship between intensity and distance

For a sphere, the surface area A is given by A = 4\pi r^2, where r is the distance from the source. Therefore, the intensity of an electromagnetic wave, I, at a distance r from the source is: I(r) = \frac{P}{4\pi r^2} As the distance from the source increases, the intensity of the wave decreases and follows an inverse square law.

Relate intensity and electric field amplitude

The intensity of an electromagnetic wave can be related to the maximum amplitude of the electric field, E, by the formula: I = \frac{1}{2} \epsilon_0 c E^2 where \epsilon_0 is the vacuum permittivity, c is the speed of light, and E is the maximum amplitude of the electric field.

Find the electric field amplitude as a function of distance

Combining the relationships for intensity vs distance and intensity vs electric field amplitude, we can relate the electric field amplitude to the distance from the isotropic source: E^2 = \frac{2I}{\epsilon_0 c} = \frac{2P}{4\pi \epsilon_0 c r^2} Taking the square root of both sides, we find the electric field amplitude: E(r) = \sqrt{\frac{P}{2\pi \epsilon_0 c r^2}} This expression shows how the maximum amplitude of the electric field of radiation from an isotropic source varies with distance from the source. It decreases as the inverse of the distance squared, following an inverse square law.

Compare with the electrostatic field of a point charge

Now, we can compare this behavior to the electrostatic field of a point charge. The electrostatic field, E_{es}, produced by a point charge Q, is given by: E_{es} = \frac{kQ}{r^2} where k is the electrostatic constant. The electrostatic field also follows the inverse square law, as E_{es} is inversely proportional to the square of the distance from the point charge, r^2. In conclusion, the maximum amplitude of the electric field of radiation from a small, isotropic source follows an inverse square law, similar to the electrostatic field of a point charge, as both decrease with the inverse square of the distance from the source or charge.

Key concepts

Inverse Square Law

Understanding the Inverse Square Law is crucial in several fields of physics, especially when dealing with point sources of light or other forms of radiation, including isotropic electromagnetic waves. Simply put, the Inverse Square Law states that the strength or intensity of an effect such as illumination or radiation is inversely proportional to the square of the distance from the source of that effect.

For instance, if we double the distance from a light source, the intensity of light at that point is not just halved; it is actually reduced to one-fourth of its original intensity. Mathematically, if an intensity is represented by I and distance by r, the Inverse Square Law can be expressed as:
\[ I \text{ is proportional to } \frac{1}{r^2} \].In the case of electromagnetic waves produced by a small, isotropic source, this law dictates that as they travel outwards, their intensity diminishes following this same principle. As such, if you move away from the source, the electromagnetic waves will decrease sharply in strength, impacting various characteristics, including the amplitude of the electric field.

Electric Field Amplitude

The electric field amplitude is a measure of the maximum strength of the electric field at a given point in space. It is crucial when discerning how potent an electromagnetic wave is at any given location. The amplitude of the electric field is symbolized as E, and it denotes the peak value we might measure if we could visualize the field itself.

The amplitude is intrinsically linked to the energy carried by the wave; higher amplitudes signify more energy. This is vital in communication technologies, where maintaining the integrity of a signal often involves managing the amplitude of the electromagnetic waves. According to the solution steps provided, the electric field amplitude of a radiating isotropic source diminishes as you move further from the source, following the relationship:
\[ E(r) = \sqrt{\frac{P}{2\pi \epsilon_0 c r^2}} \].Here, E(r) decreases with the square of the distance (r), and P represents the power of the source, c is the speed of light, and \epsilon_0 represents the vacuum permittivity.

Intensity of Electromagnetic Waves

Intensity, in the context of electromagnetic waves, refers to the power that these waves transfer per unit area. The units typically used to express intensity are watts per square meter (W/m^2). It provides a measure of how much energy is transmitted in the form of electromagnetic waves through a certain area.

Relating to our textbook solution, the intensity (I) of electromagnetic waves radiating from an isotropic source diminishes as the distance from the source increases, following the Inverse Square Law. The relationship of intensity to both distance and electric field amplitude is encapsulated by the formulas:
\[ I(r) = \frac{P}{4\pi r^2} \]and
\[ I = \frac{1}{2} \epsilon_0 c E^2 \].These equations signal that as you move away from an isotropic source, not only does the intensity fall off dramatically due to the squared distance term, but also the electric field amplitude correspondingly drops, reflecting the decrease in energy as distance increases.

Electrostatic Field

An electrostatic field is the field surrounding a stationary charge that exerts a force on other charges within its vicinity. This field is characterized by its radial nature, emanating outwards in all directions from a point charge. The strength of the electrostatic field is determined by the amount of charge and by the distance from the charge. Using the formula presented in the textbook solution, the electrostatic field (E_{es}) can be found using:
\[ E_{es} = \frac{kQ}{r^2} \].Electrostatic fields, like the electric fields of electromagnetic waves from an isotropic source, follow the Inverse Square Law. This means as you move away from the charge that creates the electrostatic field, similar to how the amplitude of the electric field decreases with distance for electromagnetic waves, the strength of the electrostatic field also diminishes proportionally to the square of the distance. Although electrostatic fields and electromagnetic fields are different phenomena, this similarity in their behavior concerning distance demonstrates the pervasiveness of the Inverse Square Law in physics.