# Confusion about inverse square law and temperature increase of an object

I know the inverse square law means that radiation intensity decreases in proportion to the radial distance from the source (I is proportional to 1/r^2).

Does this apply to the temperature increase of an object with increasing distance from the heat source? So, if an object is 10 times farther away from a heat source than an identical object, will it experience 1/100th the temperature increase? Would it be possible to calculate the temperature of a heat source from the inverse square law?

I'm confused because the Stefan-Boltzmann law says radiation emitted from a blackbody is proportional to the fourth power of temperature.

Yes, it applies, and it's not really related to the Stefan-Boltzmann law.

The energy radiated from a blackbody at temperature $T$ does indeed scale like $T^4$. Any object (blackbody or not) can absorb radiated energy, and that is the part which increases the temperature.

The inverse square law is a statement about the density of radiation (or intensity, in units of $W/m^2$) from a point source, not about either the source or receiving blackbody itself. If at a distance of one meter from a point source an object receives 1 $W/m^2$ of radiative energy, then at a distance of 2 meters the same object will receive 0.25 $W/m^2$ of energy. That's true for a monochromatic point source as well as for a blackbody, and comes exclusively from geometry.

I won't say that it maps directly on to temperature rise of a receiving object (just because that also depends on heat capacity and re-radiation and such) but a rigorous statement is that the rate of thermal energy absorption follows the inverse square law.

Regarding the question of determining the temperature of a blackbody heat source by the inverse square law, the answer is "not really". If you know the surface area of the heater, then measuring the radiated intensity at one distance is all you need. If you don't know the area, measuring the radiated intensity is only going to give you a number proportional to $AT^4$, where $A$ is the heater surface area. You can get more information about temperature by looking at the spectrum of the emitted radiation and using Wien's law: $\lambda_{max} = b/T$, where $b\approx$ 2.9 mm K and $\lambda_{max}$ is the most intense wavelength.

• Would it be accurate to say something like, if an object 10 radii away from the heat source has its temperature increase by 1 K, the same object right next to the heat source will experience a temperature increase of 100 K? Or is this inaccurate? – FeatAnalyzer May 2 '16 at 3:26
• Even assuming the two receivers are blackbodies of the same mass, cross-sectional area, specific heat, etc., there are two possible inaccuracies (that I can think of). The first would be that the final temperature depends on the heat absorbed but also the rate of re-radiation: it's a thermal-energy-in, thermal-energy-out rate balancing equation that dictates the equilibrium temperature. The second is that a blackbody heater actually can't raise an object above its own temperature, because that would violate the second law of thermodynamics. – psio May 2 '16 at 3:37
• I found a school experiment giving an inverse-square relationship between distance from a heat source and temperature change. If I'm understanding all of this, temperature increase is dictated by the inverse square law, but temperature increase could also be less taking into account these other factors, right? – FeatAnalyzer May 2 '16 at 14:02
• Yes--if you can set the situation up in the right regime, then this will be a good approximation. The"right regime" here means the receiver is way colder than the heat source, and most likely that the temperature changes are relatively small--then you'd be able to ignore my pedantic details. :) – psio May 2 '16 at 19:27
• So temperature increase is that predicted by the inverse-square law or less. OK, thanks for clearing that up. – FeatAnalyzer May 2 '16 at 21:42