I had a frustrating science discussion with a friend the other day. Somehow, our conversation had turned to the subject of cooking -- about which I know nearly nothing -- and my friend mentioned that someday he'd like to have an induction cooktop. He told me that when you place a pan on an induction cooker and turn it on, the food will begin to sizzle almost immediately.

To me that sounded like trouble. My intuition was that a fast-heating cooker would result in cooking temperatures that were way too high. I tried to explain why, but my grasp of physics and engineering is sketchy at best, and I wasn't able to get my point across.

Now that I've had time to think about it, and done some research, I think I know how to explain it. So I figured I'd post it here and maybe someone who understands the principles better will stumble on it and straighten me out.

The idea behind an induction cooktop is that the cooking pan is positioned right over an inductive electrical coil that's just under the surface of the cooktop. An oscillating current is passed through the coil, reversing direction a few thousand times a second. The current creates a magnetic field that oscillates at the same frequency. This field is just large enough to pass through the metal base of the pan, allowing the rising and collapsing magnetic fields to induce eddy currents. The metal of the pan has a natural electrical resistance, so the induced eddy currents cause the pan to heat up.

The advantage is that the cooking pan itself is the source of all the heat used to cook the food. There's no open flame or red-hot heating coil, and neither the cooktop nor the induction coil heats up. (The cooktop does come in contact with the heated pan, so it will warm up from conduction, but it's usually made of something that absorbs heat slowly.)

Imagine a pan sitting on a stove that hasn't been turned on. If the pan has been there a while, it should be at room temperature. This is not as simple a situation as it seems, because the pan isn't just at the same temperature as the room, it's also in thermal equilibrium with the room. That is, there is no net heat flow between the pan and the room, so the pan stays at a constant temperature.

This is not to say that the pan is thermally isolated from the room. The warm surface of the pan radiates heat out into the room, and the warm parts of the room radiate heat into the pan. Similarly, if any part of the pan is warmer than the air around it, heat will be conducted into the air, and vice versa. The point is that the pan and the room still exchange heat by the usual means -- radiation, conduction, convection -- it's just that the heat flowing into the pan is exactly matched by the heat flowing out of the pan.

This is a necessary condition for any object that is staying at a fixed temperature. If the heat flows in and out didn't match, the object would experience heating or cooling.

Now fire up the burner under the pan. The flame is a new source of heat that is transfered into the pan. However, simply turning on the burner does nothing to change the rate at which heat flows out of the pan. Since more heat is flowing in than out, the pan must heat up.

When the temperature of the pan rises, that affects the heat transfer between the pan and the surrounding room because a hotter pan will transfer more heat into the relatively cool room. Since the room stays at about the same temperature, the rate of heat transfer back into the pan stays about the same. (When you cook something, the kitchen probably does heat up a bit, but only by a few degrees, which is negligible compared to the cooking pan, which heats up hundreds of degrees.)

Eventually, the rate at which heat is shed to the surrounding room will be high enough to exactly offset the new heat flowing into the pan from the burners. At this point the pan will once again be in thermal equilibrium with its environment -- which now includes the burner -- and the temperature of the pan will stop changing.

The process is pretty much the same for an induction heated pan, except that instead of heat flowing in from the burner, the heat is generated within the material of the pan by induction. The pan's temperature still starts increasing, and stops only when the temperature of the pan increases to the point where heat transfer to the room is high enough to exactly offset the rate at which heat is being generated in the pan.

(I'm pretty much hand-waving away the effects of the food in the pan by assuming that there's just enough food to give us that sizzle we were talking about. Maybe it's just a few slices of bacon.)

Over a broad range of conditions, the amount by which an object changes temperature is proportional to the amount of heat energy transferred. So, for example, raising the temperature of an object by 5 degrees will require a transfer of 5 times as much heat energy as raising its temperature one degree.

Similarly, the rate at which the temperature changes is usually proportional to the rate at which energy is transfered. Any change in energy over time is called power, which is usually measured in watts. The higher the power, the more energy is transferred per second. So the less time it takes to heat an object to a given temperature, the greater the power of the heating mechanism.

For our purposes, let's assume that a typical gas stove can heat a pan to the sizzle temperature in one minute. Let's also say that when my friend says an induction cooker makes the food sizzle almost instantly, he means within 5 seconds. Since that's 1/12 the time it takes the gas burner, that means that the induction coil must produce heat in the pan at 12 times the rate at which the burner transfers heat.

Actually, since the shorter time span allows less time for the heat to flow out of the pan into the room, quicker heating is also a more efficient way to reach the same temperature. The magnitude of this effect is very dependent on the system under consideration. In the interest of keeping the math simple, I'll assume that the induction system has 10 times the power of the burner.

Now let's pull another number out of thin air and assume the burner is transfering heat at 1000 watts. Then the induction coil must transfer heat at the rate of 10,000 watts to achieve the desired heating speed.

That gets the pan to the sizzle point, but the heating process doesn't necessarily stop, since the heating only stops when the pan is in thermal equilibrium with its environment, which now includes a 10000 watt burner.

As near as I can tell from a little Googling, on a typical gas stove, a dry pan can heat to about 500°F. In other words, if the burner is transfering heat to the pan at 1000 watts, then when the pan reaches 500°F, it's transfering 1000 watts to it's environment. So how hot does the pan have to get to shed 10 times as much energy?

It's hard to say, because it depends on details of the heat transfer mechanism, which I'm not very sure of. However, since a gas burner heats a pan hot enough to cook food, I think the induction stove with 10 times the power is going to get way too hot for normal cooking.

I've since looked up a little bit of thermodynamics and learned that in the worst case, getting rid of 10 times the heat could require a full 10-fold increase in temperature difference between the pan and the room, meaning the pan wouldn't stop heating until it hit about 3000°F. That's more than enough to turn the pan into a pool of white-hot molten metal, melting its way down through the stove.

On the other hand, I think the best case would be if the process is dominated by radiative cooling, in which case the radiated energy increases as the fourth power of the the temperature, which my rough calculations show would require a temperature of only 860°F. That's still way above normal cooking temperature. If you turned the lights out, you'd see a faint red glow like a branding iron.

When I was trying to explain my concerns to my friend, I didn't present it nearly as clearly as I have here (and I realize this is still not a model of clarity). He didn't see the problem, and he kept pointing out the induction process heated the pan much more efficiently than an open flame, so not as much energy was needed.

But that misses the point. The amount of waste heat is irrelevant to the problem. In order to heat the pan 10 times faster, the induction cooker simply must pump energy into it 10 times as fast. This means that when the pan stabilizes at its equilibrium temperature, it's going to have to emit 10 times as much energy into the surrounding room. This far higher energy emission rate can only occur if the equilibrium temperature is also far higher.

My point was that something was missing from our mental model of how induction cooking worked, because our mental model predicted a top-end cooking temperature far too high for normal cooking. No engineer would design an induction cooktop to overheat food so much. There's no point to it.

We did manage to come up with a couple of theories. My friend pointed out that induction cookers only heat the bottom of the pan. This means the burners need less than 10 times the power to get to the sizzle point 10 times as fast, because they are bringing less of the pan to that temperature.

My theory was that the induced eddy currents would occur on the surface of the pan, the inside and the outside, without directly heating the interior thickness of the metal. Since the food sizzles when the surface touching it reaches the sizzle point, this again reduces the amount of mass required to be heated, thus reducing the required energy and requiring less power to reach the sizzle point quickly.

A bit of research on the web seems to indicate that both of these theories are true. Induction cookers don't heat the pan sides directly, and eddy currents do indeed hug the surface. However, it turns out that neither of those is the real solution to the overheating problem.

When heated by an induction cooker, each type of pan reacts differently, depending on its shape, size, and construction materials. To achieve optimum cooking and energy transfer, an induction cooktop has to be able to adapt its performance to each pan. Fortunately, the effectiveness of a particular frequency can be detected by measuring the rate of energy consumption of the induction coil, and the induction wave can be modulated until the energy transfer is at the desired level.

The point is that, contrary to what I had been imagining, an induction cooktop is not just a coil that blasts out a magnetic wave. The inductor's resonance circuit includes a switched power supply that is controlled by a microcontroller -- a small computer -- that continually monitors the coil current and adjusts the induction wave form from moment to moment according to a programmed formula.

Once you have a computer on board, lots of things become possible, like changing the wave if the pan is moved to a new position, or shutting off the power when the pan is removed.

Or detecting when pan overheats. In a modern induction cooktop, each induction coil has a temperature sensor that detects when the cooktop, and therefore the pan, gets too hot, at which point it cuts off the power until the pan cools.