Will a white object be cooler than a mirror when exposed to direct sunlight? Or will it be the other way around? If mirrors bounce all visible light and some of the other elements of electromagnetic spectrum, my guess is that it will be cooler than white objects–which only scatters light and not bounce them off. Do they reflect most infrared? What about UV light–which are absorbed by black objects?
[Physics] the difference between a white object and a mirror in terms of heat absorption
absorptionopticsreflectionthermodynamics
Related Solutions
If you're looking at radio waves then the mirror will have to be made of thicker metal, because as you increase the wavelength you also have to increase the thickness of the metal to get the same reflectivity. That's actually how satellite dishes work. They're basically a big curved mirror that concentrates all of the microwaves coming down from the satellite. They're often full of holes to keep the weight down, and this doesn't matter because the wavelength of the waves is larger than the holes. This is the same principle as seeing a light on in your microwave. You can see the light escaping through the door but the microwaves aren't escaping because they're too long. Once you get beyond visible light into the shorter wavelengths; ultraviolet light is easy to make mirrors for, but x-rays are very difficult. And so making x-ray telescopes is very difficult. Sometimes they do it by using a bag of gas to act like a lens rather than mirrors or by using a metal mirror but at a very grazing angle which makes the mirror very large. Mobile phone waves are at the microwave end of radio waves, and a sheet of aluminium would work nicely as a mirror for those.
Source : the naked scientists
One can see that for making lenses, materials of different refractive indices can be used based on required convergence, divervence and dimensions can be compared to those of the above described mirrors, Although there may a problem that making enormous lenses for radio waves require materials/machinery that we may not have at the moment, mirror though as you see are the big satellite dishes that we have seen many times.
You can't argue with a black object and a white object alone, as I think you partially understand in trying to build your thought experiment. You need a little bit more to define things properly. See whether the following helps.
Imagine a black object at a temperature $T_0$ and a white object also at $T_0$ inside a perfectly isolating box full of blackbody radiation at some higher temperature $T_1>T_0$ (i.e. without the black and white objects, this radiation is in thermodynamic equilibrium).
To understand exactly what would happen, you would have to describe the "colour" of your objects with emissivity curves that show emissivity as a detailed function of frequency. So your "black" and "white" would need to be defined in much more detail. You would also have to define the surface areas of the two objects and what they are made of (i.e. define their heat capacities). But all of this only effects the dynamics of how the system reaches its final state, i.e. these details only influence how the system evolves. What it evolves to is the same no matter what the details: the box would end up with everything at the same temperature such that the total system energy is, naturally, what it was at the beginning of the thought experiment. "Blacker" as opposed to "Whiter in this context roughly means "able to interact, per unit surface area, with radiation more swiftly": the blacker object's temperature will converge to that of the radiation more swiftly than does that of the whiter object, but asymptotically the white object "catches up". Blacker objects absorb more of their incident radiation its true, but they also emit more powerfully than a whiter object at the same temperature. The one concept emissivity describes the transfer in both directions. Think of emissivity as being a fractional factor applied to the Stefan-Boltzmann constant for the surface as well as being the fraction of incident light absorbed by the surface relative to a perfect blackbody radiator.
This description is altogether analogous to that of the situation where $T_0<T_1$. Begin with $T_0=T_1$, and you've got thermodynamic equilibrium from the beginning. Nothing happens, of course.
Maybe the following will help thinking about what is a really quite a complex question: it would be a fantastic last question for an undergrad thermodynamics exam BTW: You can abstract detail away by saying lets define object $A$ to be blacker than object $B$ if, when both objects are made of the same material, are the same size and shape, the temperature of $A$ converges to the final thermodynamic equilibrium temperature more swiftly than that of $B$ when they are both compared in the box-radiation-object thought experiment above.
Thinking about this now, I am not sure whether the above definition would hold for every beginning temperature of the radiation. Maybe there are pairs of surfaces whose relative blackness is different at different beginning temperatures such that $A$ is blacker than $B$ with some beginning temperature whilst the order swaps at a different beginning temperature. I think it is unlikely, but that is probably a different question altogether.
By the way, which pub do you drink in? I might come along.
Afterword on a Heater's Colour:
You ask by implication what is the best colour to paint a heater. This is not a simple question and involves the dynamics of the heater system. It's really an engineering question. I suspect in general it is better for them to be blacker rather than whiter. Here's a glimpse of the kind of factors bearing on the situation.
If you can say a heater has a constant nett input of $P$ watts, then at steady state that's going to be its output to the room, altogether regardless of its colour. There may be a materials engineering implication here: if you paint the heater whiter, and if its dominant heat transfer to the room is by radiation (rather than by convection or conduction), then it has to raise itself to a higher temperature than it would were it blacker so as to radiate $P$ watts into the room. So its materials might not be as longlasting, and it might be more of a fire hazard than it would be were it blacker.
If the heater is the hot water kind, and again if radiative transfer is significant, then the heating system has to run hotter to output power at a given level if the heater is whiter. At a given flow rate and given temperature of heating water, the heat output of heater is lower if it is whiter. You're trying to design the heater to be an "anti-insulator": you want the heat to leak out of the flow circuit in at the heater, not through the lagging on the hot water pipes outside the building channelling the water from the boiler to the heaters. If the hot water pipes leak heat in the same room, then that's no problem.
Recall the quartic dependence of the Stefan Boltzmann law. At room temperatures with a low temperature heater (the hot water kind) $\sigma\,T^4$ is likely to be pretty small compared with other heat transfer mechanisms, in contrast to my idealised scenarios above. So the heater's colour is likely to be pretty irrelevant.
Best Answer
The answer to this question depends on various aspects. For instance when you say white object, do you mean perfectly white? or white with respect to only visible light? Same happens for a mirror too.
The most direct way to look into the problem would be following. Mirrors are nothing but extremely fine and optically flat white surfaces at the back of glass. Usually its silver coating. Silver when left alone as a lump can act as a white object too. Now we can reformulate the question by asking: Will a fine surface of silver absorb less heat than a lump of silver? The answer is their absorption co-efficient is same. So heat absorbed per unit mass will be same.
But mirror looks more brighter as if it reflects more light than white object. The answer to this is that usually white objects do not have flat surface. So the reflection is not regular unlike mirror.
P.S. If you consider the glass of mirror through which light has to pass, you'll find that some light is absorbed their. So reflectivity of mirror is actually less.