# Microwave oven and heating my sons milk

I heat up my sons milk in the microwave...

I use the same volume all the time and always punch in 42 seconds (on high) and his milk comes out the perfect temp every time.

However, sometimes he wants "mo milk" and I just want to give him a half bottle. So, I was filling up the coffee cup with half as much milk and I was cutting the micowave time from 42 seconds to 21 seconds. Seemed like a logical thing to try. But I am finding the milk to be way too hot. It does not seem directly proportional. (No I have not taken out a thermometer to prove it, but I'm pretty sure this is the truth)

My question is: How can I figure out how many seconds to heat my sons milk so that a half glass of milk is the same temperature as when I microwave a full glass?

Thanks.

I have a question, and an answer, for you.

The question: how do you determine that the milk in the half full cup is "too hot"? Are you feeling the cup, or the milk? I ask because some ceramics will absorb microwave power; this might make the cup "feel" hot when there is not much other material around to absorb the power (like the milk in the cup).

The proposed method:

I am assuming that your hand is a good tool for measuring how hot the milk is, and that you have an accurate scale (for measuring amount of milk).

Weigh the empty mug. Call its weight $m$.

Add milk until it is half full. Call new weight $M$.

Run the microwave for 21 seconds, observe milk is too hot. Now add cold milk slowly, until the milk feels "right". Weigh the mug, call new weight $M'$.

The time to heat half a cup of milk will be approximately

$$21 \times \frac{M-m}{M'-m}$$

Repeat the experiment with the shorter time, and confirm it is correct. If necessary, make another adjustment.

The surprising thing is that if the heat capacity of the mug was a significant factor in this, you would expect that 21 seconds was not enough to heat half a mug.