Why did the microwave oven only heat my coffee half as much as expected?

A sticker on my microwave oven states its output effect to be $750\ \mathrm W$, which is $180$ calories per second. This means that heating $250\ \mathrm g$ of water by one degree Celsius would take $250/180 = 1.4\ \mathrm s$.

Now, my cup of $250\ \mathrm{cl}$ coffee had cooled to $35$ degrees, so I figured heating it to $45$ degrees would take $14$ seconds. But the actual temperature after that time was only $40$ degrees.

I did not measure the coffee volume exactly, and the thermometer probably has an error margin, but still the $50\%$ deviation makes me think that I've failed to take something into account. The porcelain cup should not have been heated much, and $14\, \mathrm{s}$ is probably too short for much heat to transfer from the coffee to the cup.

• the rating is maximum...It might now be running at that rating all the time... – Vineet Menon Apr 10 '12 at 15:30

Your microwave creates a standing electromagnetic wave inside itself but it doesn't consume much energy to create this wave, so if the oven is empty it will only consume a small amount of power - a perfect microwave oven would consume no power when empty. Power is only consumed when you put something in the oven that absorbs energy from the standing wave. The oven needs to take energy from the mains to replace the energy absorbed by whatever's inside it.

The rate at which something inside the oven absorbs energy depends on what you put in. For example if you put in a cup of cooking oil it will absorb energy slowly and heat up slowly (I wouldn't try this at home as traces of water in the oil can boil explosively!!).

The 750W rating of the oven doesn't mean it pours 750W into whatever is inside it; it means that's the maximum amount of power it can pour in. The actual power absorbed will typically be less than this and possibly much less.

It would be an interesting experiment to try heating two (or more) mugs of coffee at the same time. I would bet you'll find the total energy absorbed increases as you put more mugs in, up to the 750W limit.

Later:

Prompted by Anna I have done the experiment. I used two identical coffee mugs containing 400g of water each. I first heated just one mug and measured the temperature rise every ten seconds from about 10C up to about 25C - I didn't want to go higher because you have to start worrying about heat loss to the mug and air.

I then replaced the water and heated the two mugs together, spaced as widely apart as possible, and again graphed the temperature as a function of time again up to about 25C. The results? Well my oven is rated at 600W and with one mug I measured the rate of temperature rise to be 481W (plus or minus a few percent). With two mugs the rate of temperature rise was 530W.

Now I'll just post the results up to the arxiv :-)

• Hmm. I am very fond of jacket potatoes in the microwave oven. One potato 5 minutes is to my taste, two potatoes need 10 minutes. It is only when there are a lot of people, for example 6 potatoes, that the time is not multiplicative. – anna v Apr 10 '12 at 9:59
• I've updated the original post with some experimental results. – John Rennie Apr 10 '12 at 11:18
• By the way, is there a method for approximating how much energy the water will absorb affect below the max effect, without actually measuring? – Per Apr 10 '12 at 12:00
• Microwave ovens heat food by dielectric heating, i.e. the energy is absorbed by electric dipoles in the food. So the more polar the food the better it will absorb the microwaves. That's why water, which is full of polar OH bonds, heats up faster than oil. However predicting the absorption from first principles would be very difficult. To get more than a rough guess you'd have to measure the absorption. There will also be an effect of position in the oven. The standing waves mean that the microwave intensity isn't the same everywhere in the oven. – John Rennie Apr 10 '12 at 13:15
• Another factor is that the time the microwave is 'on' - light on, turntable going round - isn't the time the microwaves are on. When we used a domestic microwave in the lab we found that the uwave power didn't come on for 5-10secs after the unit started so it's hard to time short bursts. – Martin Beckett Apr 16 '12 at 16:25

The 750 watt output on your oven is usually measured using an IEC standard method that requires using one liter of water at 10 C. It is an averaged number, so your oven can actually have a much lower power,(they usually do - I'm just publishing a paper on this) by as much as 20 % or more. Second, as the water load becomes smaller, the efficiency of capture of the microwaves decreases, so the water or coffee may only see 100 or 200 watts of microwaves - a good thing, otherwise small loads would heat much too fast. Also, note that water heats most efficiently at lower temperatures; as the water temperature increases the efficiency decreases because the resonant frequency of the water increases, but the microwave frequency stays the same. These are some of the major reasons. This is all well known microwave oven physics.

• It may be well known, but you haven't explained the 50%, you only explained a little more than 20%. – Ron Maimon Apr 17 '12 at 0:17

You probably managed to get only one hotspot of maximum wave amplitude hitting your coffee while two is more usual. Putting your cup off-centre on the rotating dish would probably get the more average heating you expected.