Inverse square law, assumptions in its application.

The idea

The inverse square law says a quantity such as intensity or radiation flux falls in proportion to 1/r², where r is distance from a point source. Doubling r quarters the value; tripling r reduces it to one ninth. In astrophysics it links apparent brightness b, luminosity L and distance d by b = L/(4πd²). The law holds only under stated assumptions: the source radiates uniformly in all directions, acts as a point source, and the intervening space is empty so no absorption or scattering occurs. For a star, the point-source assumption is acceptable when d is much larger than the star's radius.

What earns the marks

• States that intensity or flux is inversely proportional to the square of the distance from a point source, written as I ∝ 1/r² or b = L/(4πd²).
• Applies the law quantitatively, for example showing that moving from distance r to 2r reduces received intensity to one quarter of its original value.
• Identifies the point-source assumption: the source must be small compared with the distance at which it is observed, so that rays are effectively radial.
• Identifies the isotropic-emission assumption: power is radiated uniformly in all directions, so the same total power passes through every sphere of radius r.
• Explains that the intervening medium must be transparent, with no absorption or scattering, so that all emitted radiation reaches the observer.
• Uses the law to compare two situations by forming a ratio, such as b₁/b₂ = (d₂/d₁)², avoiding unnecessary substitution of constants.
• Recognises that the law is a limiting case and states a realistic condition under which it fails, such as observing a star from within its own planetary system or through dense interstellar dust.

Common mistakes and how to put them right

• Treating intensity as inversely proportional to distance rather than to distance squared.

  Put it right: Write I ∝ 1/r² and check that doubling distance quarters the value.

• Applying the law to an extended source at close range without commenting on the point-source assumption.

  Put it right: State that the source must be far enough away that its angular size is negligible.

• Ignoring absorption or scattering by dust and gas and assuming all emitted power arrives.

  Put it right: State that the medium must be transparent for the inverse square law to apply directly.

• Confusing luminosity, a property of the source, with intensity or flux, which depends on distance.

  Put it right: Define each symbol clearly before substituting.

Exam tips

• Write the relationship as a proportionality first, then convert to an equation with the 4πd² factor, so the geometry behind the law is visible.
• When comparing two distances, use ratios and cancel constants; this reduces arithmetic and shows understanding of the inverse square behaviour.
• Always name the assumptions explicitly when a question asks why a calculated value may be unreliable, linking each assumption to a physical situation.
• Check units and significant figures at the end, and ensure distance is in metres before squaring.