Energy and Momentum Attenuation Law of Spherical Waves

A spherical wave is produced when a point source emits radiation that propagates outward in all directions. In the far‑field region—where the observation point is many wavelengths away from the source—the electric field E and magnetic field B of an electromagnetic spherical wave are radially symmetric. Their magnitudes depend only on the radial distance (r) from the source, not on the angular coordinates.

For a point source located at the origin, the amplitude of the electric field in the far field follows

[
E(r) \approx \frac{E_0}{r},
]

where (E_0) encapsulates the source strength. Because the ratio (E/B) equals the speed of light (c) in vacuum, the magnetic field obeys the same (1/r) decay:

[
B(r) \approx \frac{B_0}{r}.
]

This (1/r) dependence is the foundation for all subsequent energy and momentum attenuation analyses.

Energy Flux and the Poynting Vector

The flow of electromagnetic energy is described by the Poynting vector (\mathbf{S}):

[
\mathbf{S} = \frac{1}{\mu_0},(\mathbf{E}\times\mathbf{B}),
]

where (\mu_0) is the permeability of free space. For a spherical wave, (\mathbf{E}) and (\mathbf{B}) are both perpendicular to the radial direction (\hat{\mathbf{r}}), and their cross product points outward along (\hat{\mathbf{r}}). Thus (\mathbf{S}) is purely radial and its magnitude represents the power per unit area carried by the wave.

The intensity (I) is the time‑averaged magnitude of (\mathbf{S}):

[
I = \langle |\mathbf{S}| \rangle.
]

Because both (E) and (B) fall off as (1/r), the intensity diminishes as

[
I(r) \propto \frac{1}{r^2}.
]

This is the celebrated inverse‑square law: the power delivered to a unit area decreases with the square of the distance from the source. Physically, as the wavefront expands, the same total power is spread over a sphere of surface area (4\pi r^2), forcing the local intensity to drop as (1/r^2).

Momentum Density and Its Decay

Electromagnetic fields carry not only energy but also momentum. The momentum density (\mathbf{g}) is directly linked to the Poynting vector:

[
\mathbf{g} = \frac{\mathbf{S}}{c^2}.
]

Consequently, the radial component of momentum density follows the same (1/r^2) law as the intensity. Two important consequences follow:

  • Local momentum density at any point in space decreases as (1/r^2).

  • Total momentum flux through a spherical surface of radius (r) remains constant in a lossless medium. The integral

    [
    \mathbf{P}{\text{total}} = \int{A} \mathbf{g},c,dA
    ]

    yields a result independent of (r) because the (1/r^2) decay of (\mathbf{g}) cancels the (r^2) factor from the surface area.

Thus, while the local density of momentum diminishes, the overall momentum carried by the wave is conserved.

Conservation of Energy and Momentum

Energy Conservation

The Poynting theorem states that the rate of change of electromagnetic energy within a volume equals the net outward energy flux minus any work done on charges. For a spherical wave in free space, the total power crossing a sphere of radius (r) is

[
P = I(r),4\pi r^2 = \text{constant}.
]

Even though the intensity (I(r)) falls off as (1/r^2), the product with the surface area remains unchanged, confirming that energy is conserved during propagation.

Momentum Conservation and Radiation Pressure

When a spherical wave impinges on a surface, it exerts a pressure due to the transfer of momentum. The radiation pressure (p_{\text{rad}}) depends on whether the surface absorbs or reflects the wave:

  • Perfect absorption: (p_{\text{rad}} = \dfrac{I}{c}).
  • Perfect reflection: (p_{\text{rad}} = \dfrac{2I}{c}).

Because (I \propto 1/r^2), the radiation pressure also decays rapidly with distance. This principle underlies phenomena such as the pressure exerted by sunlight on spacecraft sails.

Practical Applications

The inverse‑square attenuation of energy and momentum has wide‑ranging implications across science and engineering.

  • Astronomy: Light from stars travels as spherical waves. By measuring the received intensity (I_{\text{obs}}) and knowing the star’s intrinsic luminosity (L), astronomers compute the distance (r) via (I_{\text{obs}} = L/(4\pi r^2)). This method underpins the cosmic distance ladder.
  • Radio and Microwave Communication: Free‑space path loss follows the (1/r^2) law. To compensate, high‑gain antennas (e.g., parabolic dishes) collect more of the expanding wavefront, effectively increasing the received power.
  • Sonar and Acoustic Sensing: Although sound is a mechanical wave, in a homogeneous medium its spherical propagation also obeys the inverse‑square law. This dictates the range and resolution limits of underwater detection systems.
  • Optical Trapping and Radiation Pressure Experiments: Precise control of momentum transfer to microscopic particles relies on the predictable decay of radiation pressure with distance.

Summary

Spherical waves from a point source exhibit a characteristic (1/r) decay in field amplitude, which translates into a (1/r^2) reduction of both intensity and momentum density. This inverse‑square behavior ensures that, despite local attenuation, the total energy and momentum remain conserved as the wave propagates through free space. Understanding these attenuation laws is essential for interpreting astronomical observations, designing long‑range communication links, and exploiting radiation pressure in advanced technologies.