# What would a graph of temperature increase of a cup of water in a microwave look like?

My lunch had been in the microwave for a minute or so, and I was wondering if I took it out 10 seconds early, would the amount of temperature it increased in that 10 seconds be more significant, less significant, or the same as the temperature increase over the previous 10 seconds. That is, is the temperature increase linear, exponential, or logarithmic? Or something else?

My friend suggested it would depend a lot on whether it was covered or not due to the steam build-up, but I don't know.

Here is a link to a study comparing heating water in a microwave to heating water in a conventional oven.
Depending on the power of the microwave, the volume of the water, and time it's placed inside, the temperature will vary approximately linearly with time until either the system reaches equilibrium (for low power microwaves and large volumes of water) or the water starts to change phase (i.e. begins boiling), at which point the temperature will start to level off.

Here's the relevant graph:

Just to add an explanation to sh37211's good empirical answer: the microwave transfers energy to the water at a constant rate. For a microwave rated 1,300 Watts and a typical 50% efficiency$^1$ that means a constant 650 Watts of heat energy delivered to the water. To calculate the change in temperature $(\Delta T)$ of a mass of material $(m)$ based on how much heat energy $(Q)$ was added, you use the material's heat capacity $(C)$:

$\Delta T= \frac{Q}{mC}$

The heat capacity of liquid water is just a constant, $4.18\ J/g\cdot K$. As you can see, $\Delta T \propto Q$. Since the $Q$ provided by the microwave is constant over time, so is the $\Delta T$. Thus one would expect a linear relationship between the temperature of the water and time.

$^1$ see page 2; efficiency is defined as the power delivered to water divided by the power consumed by microwave oven