I felt this needed a physics answer.
tl;dr: It's unlikely but not impossible - we have to make some unrealistic assumptions to make it work for a sensible size of potato
Consider a homogeneous spherical potato in a vacuum*:
According to Wikipedia potatoes are 79% water, 17% carbohydrate (mostly starch), 2% protein and 2% fibre. Figures I've seen elsewhere vary by a few percent, so we can simplify to 80% water and 20% starch (the thermal properties of protein-and fibre-rich foods are close to those of pure starch so we can treat them as the same).
Water has a specific heat capacity of 4.3 kJ/kg/°C, and starch 1.75 kJ/kg/°C. That means that our model potato has a specific heat capacity of 3.7 kJ/kg/°C (a real measured value is a little lower at 3.43 kJ/kg/°C, but as we will see, water dominates the results).
First we consider the heat energy required to get the potato up to the boiling point of water (100°C), assuming a starting temperature of 20°C (room temperature), and a potato weighing 0.2 kg (just under half a pound, so a small jacket potato). 0.2×80×3.7 = 59.2 kJ. 1 kJ is 1 kW for 1 s, so a 1 kW microwave, 100% efficient, would heat this potato to boiling point in about 1 minute. This is considerably quicker than realistic for several reasons, and this is something we're familiar with:
- The microwaves don't penetrate to the centre so much of the potato is heated only by conduction.
- While this is going on the hot surface is losing heat to its surroundings by convection, conduction, and radiation.
- Microwave ovens aren't 100% efficient at delivering their stated power into food, though it's surprisingly hard to get figures on losses.
Once the temperature of the potato is 100°C, it won't keep on rising; instead the heat will go into evaporating the water in it. The latent heat of vapourisation of water is 2300 kJ/kg or 460 kJ for our potato. This is near enough another 8 minutes (total 9 so far) at 1 kW, before we can start to get above 100°C by boiling off all the water (I assume the potato is isothermal, which of course isn't true, but thew timescales are long enough for significant heat flow).
Once we've boiled off all the water we can start heating the remaining dried potato (starch). We only have 20% of the mass left (i.e. 0.04 kg of starch, the specific heat capacity of which is 1.75 kJ/kg/°C). Red heat is at least around 700°C (for steel, but the potato would char to carbon at lower temperatures and carbon should be close enough to steel). The extra 600°C rise would take another 600×0.04×1.75 = 42 kJ of energy, or 42 seconds.
So it's just about possible to get a 200 g potato red hot in a microwave in 10 minutes, but only if we:
- neglect all heat losses (except those due to driving off water), which would be significant. I won't go into detail but the potato would have a surface area of 154 cm2 and would radiate only 12 W net at 100°C but 168 W at 400°C (just below the autoignition temperature, and a significant fraction of the input power). The plate/turntable takes away some heat by conduction, and we can;t really neglect convection (but it's hard, hence the assumption of a vacuum).
- assume 100% microwave efficiency.
On the other hand we assume no energy is released by combustion, and if we can get the dry starch to its autoignition temperature of 410°C (Wikipedia) it will burn, releasing quite a lot of heat (the heat of combustion of starch is 17.5 mJ/kg, but much of this is lost to the potato).
Another point that makes burning more likely is that we don't actually need to get to zero water in the whole potato before some of it can start burning. This would mean that more of the heat of burning some potato is useful in heating the rest, and would be of particular interest if the turntable of the microwave was jammed or there was a hotspot that received considerably more power than the rest. A similar effect would be caused by conductive (e.g. metal) particles on the potato, as they'd cause a small region to get very hot. These don't really fit with the even burning shown in the photo.
Those who are interested in the physics may wish to look at The Thermal Properties of Potato, T. Yamada 1970. It's in Japanese with an English abstract but the figures give clear results for the specific heat capacity and thermal conductivity. A 1-dimensional thermal model would be an interesting undergrad physics problem.
* not strictly required but for once a sensible simplification - and we physicists so rarely get the chance to follow this stereotype.