If you look at the spec sheet, there are graphs with the magnitude of current thru the device given for certain values of gate to source voltage. At 5 volts for your device you will notice that no matter how you change the voltage at the drain, the total current does not change (it is constant). In this situation if you do not change the load resistance and the total current is constant the voltage from source to drain must change because the voltage across the load will not. This increased voltage along with the current (IR drop) must result in a increased power dissipation (P=IE) in the device.Thanks for the analysis, marc. It's now clear that FQP33N10 is the wrong part to use for my device, but out of curiosity, which graph were you looking at (trying to understand what you mean exactly by going linear)?
Also, do you mean I should not use any resistor between Arduino PWM output and the gate?
I looked at the specs of this FET, but accordingly to the power and temp estimations suggested by ronv, it will not survive the task
Also you may want to look up the term 'load line'
Another way to look at it is to consider a class A amplifier. It has a no ac input signal output of 1/2 of its power supply at a certain current at the point where the collector (or drain) are connected to the load.
That 1/2 of the power supply at a certain current is dissipated in the transistor in the class A (linear) amplifier, whereas in a switching mode amplifier (as in pulse width modulation) the emitter to collector (source to drain) voltage is very small, dissipating much less power in the device.
As p = i.e. and e = ir the power in terms of resistance and current is ids squared times rds. This action makes linear amplifiers work, but when you just need to either have it on or off you do not want to have a voltage drop inside the device, you want all the voltage to be developed across the load.
I also think that you may want to consider using a TO5 package and a serious heat sink. I do not think TO220 packages are going to cut it.
As far as using a resistor in series with output of the ardunio and the gate, there is no reason to, the important thing is to make sure there is a pulldown resistor from the gate to the drain. The gate to drain resistance is essentially infinite and looks like a 0.001 ufd capacitor. What you wind up with if you use a series resistance from the pwm output is a RC time delay circuit.
Look up the term 'integrator'
Marc