Question

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 from a small isotropic source is inversely proportional to the distance from the source (E ∝ 1/r). On the other hand, the electrostatic field of a point charge decreases with the square of the distance (E ∝ 1/r^2). This means that the electric field amplitude of an isotropic source decreases more slowly with distance than the electrostatic field of a point charge.

Step-by-step solution

a) Variation of maximum amplitude with distance

An isotropic source is a source that emits radiation equally in all directions. For an isotropic source, the electric field amplitude decreases as we move away from the source, as the energy is distributed over a larger surface area. This relationship is inversely proportional to the distance from the source. The power radiated by an isotropic source can be given as P = 4 * pi * r^2 * S, where P is the power, r is the distance from the source, and S is the intensity. The intensity is related to the electric field E as S = (1/2) * ε0 * c * E^2, where ε0 is the vacuum permittivity and c is the speed of light in vacuum. Combining these equations, we can find the maximum amplitude of the electric field as a function of distance: E = sqrt((2 * P) / (4 * pi * r^2 * ε0 * c)) As we can see, the electric field amplitude E is inversely proportional to the distance r: E ∝ 1/r.

b) Comparison with the electrostatic field of a point charge

In the case of an electrostatic field generated by a point charge, the field is given by Coulomb's law: E = (k * q) / r^2 where E is the electrostatic field, k is the electrostatic constant, q is the charge, and r is the distance from the charge. Comparing the two results, the electric field of an isotropic electromagnetic source decreases with the distance (E ∝ 1/r) while the electrostatic field of a point charge decreases with the square of the distance (E ∝ 1/r^2). This means that the amplitude of the electric field of radiation from a small isotropic source decreases more slowly with distance than the electrostatic field of a point charge.

Key concepts

Electric Field Amplitude

Understanding the behavior of an electric field's amplitude is crucial when studying electromagnetic waves. When dealing with a small, isotropic electromagnetic source, the electric field amplitude is a measure of the strength of the electric field at a given point in space.

The important thing to remember is that the electric field amplitude (\( E \) of radiation from an isotropic source decreases as the inverse of the distance from the source, following the relationship: \[ E = \frac{\sqrt{\frac{2P}{4\pi \epsilon_0 c}}}{r} \] where \( P \) denotes the power of the source, \( r \) is the distance from the source, \( \epsilon_0 \) stands for the vacuum permittivity, and \( c \) is the speed of light.

This decrease is understood by visualizing how the radiated energy spreads out over a spherical surface, which increases in area with the square of the distance (\( 4 \pi r^2 \) area of a sphere). Thus, as one moves away from the source, at a greater distance, the surface area over which the energy is spread becomes larger, causing a reduction in the electric field amplitude experienced.

Radiation Intensity

Radiation intensity is used to describe the distribution of power over a surface or the amount of energy flowing through a given area. For an isotropic source emitting electromagnetic waves, this intensity at a distance from the source helps us quantify how much power is received per unit surface area.

As the electromagnetic waves propagate, the radiation intensity is given by the formula: \[ S = \frac{1}{2} \epsilon_0 c E^2 \] where \( S \) is the radiation intensity, \( \epsilon_0 \) represents the permittivity of free space, \( c \) is the speed of light, and \( E \) is the electric field amplitude, which as we discussed earlier, decreases with distance.

This relationship highlights that as the distance from the source increases, the intensity of the radiation follows the inverse-square law, diminishing as the square of the distance from the source. This is vital when comprehending how electromagnetic waves lose their strength over distance; something essential in applications ranging from radio broadcasting to the safety standards for radiation exposure.

Coulomb's Law

Coulomb's Law is a foundational principle in electrostatics, describing the force between two point charges. It states that the electric field (\( E \) due to a point charge is directly proportional to the charge itself (\( q \) and inversely proportional to the square of the distance (\( r \) from the charge: \[ E = \frac{kq}{r^2} \] where \( k \) is Coulomb's constant.

This inverse-square law is distinct from the inversely proportional relationship we see with an isotropic source's electric field amplitude. For the static case of point charges, the field strength diminishes more rapidly - with the square of the distance - compared to the linear diminishment of an isotropic source's radiation field.

Understanding Coulomb's law is fundamental to not only grasping electrostatics but also in seeing the difference in how field strengths vary between static electric fields and those associated with electromagnetic radiation. As students explore various electromagnetic phenomena, distinguishing between these two inverse relationships provides clearer insights into the principles governing electric fields and helps in understanding their implications in the physical world.