Why is there an alpha (α) in E ∝ P / d²?

The formula E ∝ P / d² (illuminance proportional to power divided by distance squared) is the standard simplified form of the inverse square law for light from a point source, and it is not wrong — it's just an approximation or shorthand that omits some constants for simplicity when we only care about proportionality.

📐 The Full Equation

The precise relationship for illuminance E (in lux) from a point source of radiant power P (in watts) is:

E = (P × η × cos θ) / (4π d²)

Or more commonly in photometric terms (using luminous power):

E = I / d²

where I is the luminous intensity (candela = lumen per steradian) in the direction of the surface.

Components of the Full Radiometric Form

E = irradiance/illuminance (W/m² or lux)

P = total radiated power (watts or lumens)

4π comes from the surface area of a sphere (4πr²) — light spreads uniformly over the sphere's surface, so intensity drops as 1/r²

🔍 Why Alpha (α) Appears

In some textbooks, engineering contexts, or non-ideal cases, you see:

E = α P / (4π d²)

Here α (alpha) is not a mistake — it is a multiplicative constant that accounts for real-world deviations from the ideal point-source assumption.

Typical Meanings of α

Absorption/transmission factor (0 ≤ α ≤ 1) — fraction of power that actually reaches the surface after atmospheric absorption, glass/filters, etc.
Cosine factor (cos θ) — when the surface is not perpendicular to the rays (Lambert's cosine law). For θ = 0° (normal incidence), cos θ = 1, so α = 1.
Efficiency or luminous efficacy factor — if P is electrical power (watts) not luminous power (lumens), α = luminous efficacy (lm/W) × optical efficiency.
Geometry/view factor — for non-point sources, extended sources, or non-uniform beams, α adjusts for how much of the total power actually illuminates the surface (solid angle subtended).
Reflectance/specularity correction — in some contexts (e.g. machine vision or material testing), α includes surface reflectance if measuring reflected light.

So E ∝ P / d² is correct when α = constant (often assumed = 1 for ideal cases), but E = α P / (4π d²) is the more general, accurate form when any of those real-world factors are non-unity.

💡 Why the 4π is Often Dropped

In many practical lighting/engineering contexts (especially photography, stage lighting, or quick estimates), people write E ∝ 1/d² or E = I / d² (where I is intensity in candela), and drop the 4π because:

They are using luminous intensity I (cd) rather than total power P (lm or W).
The 4π is absorbed into the definition of candela (1 cd = 1 lm/sr, and full sphere = 4π sr).
For relative comparisons ("twice the distance = quarter the light"), the constant cancels out anyway.

So E ∝ P / d² is a correct proportionality — just not the full equation with units.

✅ Conclusion

Is E ∝ P / d² wrong?

No — E ∝ P / d² is correct and widely used for proportionality and quick estimates.

Is it incomplete? Yes, if you want absolute values with units — then you need the full form E = (P / 4π d²) (radiometric) or E = I / d² (photometric), possibly with α for real-world corrections.

α appears when authors want to highlight non-ideal factors (absorption, angle, efficiency, etc.) — it's not an error, just extra precision.

In the context of courtroom lighting / police light calculations, E ∝ P / d² was used correctly as a proportionality to compare relative brightness — the constant factors (4π, cos θ, efficiency) cancel out when comparing different sources or distances. So no mistake there!