Inverse-Square Law: Isotropic vs Anisotropic Radiation

[GRB 080319B]

If the intensity of electromagnetic radiation (I) is equal to the power emitted (P) divided by the area of the sphere (4pi(r^2)) that it radiates, why is the inverse-square law 1/(r^2) instead of P/(4pi(r^2))? Shouldn't the radiation emitted propagate isotropically in all directions (sphere) instead of anisotropically (cube)? I apologize for the notation; I don't know how to write in latex. Thank you.

 

[rbj]

the $4 \pi$ factor can just be folded into the "k" or "G" factor we see in inverse-square laws. either way, they're inverse-square.

but, i agree with your sentiments, which is why i think that more natural Planck Units would be those that normalize $4 \pi G$ instead of just $G$ and it would be better to choose a natural unit of charge that would normalize $\epsilon_0$ instead of normalizing $4 \pi \epsilon_0$.

 

[sir-pinski]

The inverse-square law is a fundamental principle in physics that describes the relationship between the intensity of radiation and the distance from the source. It states that the intensity of radiation decreases in proportion to the square of the distance from the source. This law applies to all types of radiation, including electromagnetic radiation.

The reason why the inverse-square law is 1/(r^2) instead of P/(4pi(r^2)) is because of the way radiation propagates. Radiation does not spread out uniformly in all directions, like a sphere. Instead, it spreads out in a specific pattern, depending on the source. This pattern is known as the radiation pattern.

For example, a point source of radiation will emit radiation in all directions, but the intensity will be greatest in the direction of the strongest emission. This creates an anisotropic radiation pattern, where the intensity is not the same in all directions. This is why the inverse-square law is not simply P/(4pi(r^2)), as the intensity is not evenly distributed in a sphere.

On the other hand, if the radiation source is isotropic, meaning it emits radiation equally in all directions, then the inverse-square law would be P/(4pi(r^2)). This is because the intensity of radiation would be the same at any distance from the source in all directions, creating a spherical radiation pattern.

In summary, the inverse-square law is a result of the specific radiation pattern of the source, which may be isotropic or anisotropic. Therefore, the notation 1/(r^2) reflects the anisotropic nature of radiation, while P/(4pi(r^2)) would apply to isotropic radiation.