"Very interesting question! There are a few things that we need to consider here, so we should take it step by step. First, we need to understand what is making the boxes heat up, which we know to be energy from the sun. However, these two boxes will experience different types of heat transfer from the sun to the air inside.

The sunlight hitting a clear glass box will heat the air inside by a heat transfer mechanism known as radiation. This is the transfer of energy from the radiation coming from the sun to the air molecules trapped in the box. The air surrounding this box will also be experiencing this radiation (maybe even more depending on whether or not the glass allows all radiation to pass through it!) but it won't heat up like the air that is trapped. That is because the air in the environment can transfer that energy to its surroundings through a different type of heat transfer known as convection. This happens when the air that has been heated up expands, which lowers the local pressure of this air pocket. The surrounding cooler air is denser, so the air molecules will move themselves from the area of high pressure to the area of low pressure. This local air movement also serves to transfer that heat from the hot air pocket to the cooler air pockets, thus dissipating that energy.

However, the air inside the glass box doesn't have that freedom to move. As it begins to heat up inside the box, there will be small pockets of hot and cool air that move and mix, but as the air continues to dissipate that energy through convection, all of the air in the box will see a temperature increase because there is no cooler air from farther away that is able to come in the box and mix with the hot air. What's more, the glass walls are likely good thermal insulators, meaning heat doesn't transfer quickly through the glass to the outside. The rate at which this temperature increases will depend on a few factors, namely the transparency of the glass box to the sun's radiation, and how quickly solar radiation can heat up the volume of air inside the box.

The metal box will behave a little differently. Radiation from the sun won't be able to directly hit the air molecules in the box, so the trapped air can't heat up through radiation. Rather, the radiation from the sun will hit the metal walls of the box, which will heat the walls themselves. It's likely that these walls will not absorb all of this energy, because metal will reflect much of this radiation. These walls are made of steel, which is much better at transferring heat through itself than glass is. This is a third type of heat transfer known as conduction, which is the act of thermal energy being transferred between a high temperature area and a low temperature area by way of the material itself. Once these walls heat up through radiation and conduction, the air that is near the walls on the inside of the box will transfer this energy from the wall to the trapped air again through convection.

What happens next will look very similar to what's happening in the glass box, with cold air pockets moving into hot air pockets and dissipating this thermal energy through the trapped air, eventually raising the temperature of the entire volume. It is often assumed that, if the steel walls are thin enough, the time it takes for conduction to equalize the temperature across the thickness of the metal walls is negligible, so the rate at which this temperature increases will instead depend on how quickly the trapped air transfers this heat from the walls to itself through convection.

Considering these two heat transfer methods that are present with these two boxes, it becomes clear that the glass box will heat up faster. Both boxes are experiencing convection within them to gradually raise the temperature of the volume of trapped air, but the glass box allows for higher energy solar radiation to flood in and directly heat the full volume of trapped air, while the metal box relies on the solar radiation to first heat the metal box, and then the metal box to transfer that lower energy heat to the inside through the pockets of air that are closest to the internal walls.

The two features at play here are the greenhouse effect that can occur for the glass box vs. the higher thermal conductivity of steel (about 100-5,000 times that of glass and plastic depending on the specific type- but all we care about for now is that it is “much higher”).

The greenhouse effect occurs when shorter, visible wavelengths of light pass through a visibly transparent medium (glass in this case) and then are adsorbed by objects. This heats the objects, which eventually re-radiate that heat but at longer wavelengths in the infrared range. These do not pass readily through the glass, and thus the heat is trapped in the box. However, if your glass box is empty (no objects to absorb light) and positioned such that light can pass right through it, this effect will not occur. So my initial thought is that the answer to your question depends on what is in your boxes. If they are empty, the steel box will get hotter as it has a much higher thermal conductivity. If the boxes had a reasonable number of objects in them, I would imagine the greenhouse effect would dominate, and the glass/plastic box would be hotter (imagine the temperature difference between the trunk vs passenger cab of a car on a sunny day).

Hi Peter, this is a good question! This property that will help you answer this question is the thermal conductivity of your material. If a material is more thermally conductive, it will absorb and distribute heat more evenly, more quickly. As it heats up, it will also radiate more heat both inside and outside of the box. This will raise the internal temperature of the box. Metals are some of the most thermally conductive materials that exist. As such, I believe that, for boxes of the same size, the sealed steel box will heat up internally faster than the glass/flexiglass one.

That depends on what is inside of the box.

The steel box will absorb some visible light that the glass box will not, but the glass box will absorb more infrared light. If the glass box is empty, then it will absorb less energy in total because the visible light will pass through it. Thus, the steel box will be warmer, since it will reach a higher equilibrium temperature before the heat that it radiates equals what it absorbs.

Place something in the glass box that is highly absorptive, however, and now the glass box will have a higher equilibrium temperature, because now it absorbs all of the sun's energy, and also the absorptive material inside will not be able to get its heat out, thanks to the greenhouse effect (infrared light emitted inside will be absorbed by the walls of the box).

Yes! A good experiment. What color is the steel box? What color will the transparent box be sitting on? The steel will conduct heat fastest. The Plexiglas and glass will conduct heat differently from each other, and the glass's heat conductivity will be closer to the steel's - the Plexiglas's heat conductivity will be farther from the steel's, based on how the 2 transparent materials feel to the touch. I'll let you do the experiment.