I'm home and I've taken a photo of the 1/4 watt, "Supposed" 1/2 watt and the bigger 1/2 watt and I've taken a photo next to a ruler and it seems that the 1/4 and the small 1/4 are the same size but I did find the pack that the small 1/2 watt were in and it said it was "mini" could that be the reason it's so small?
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The power rating on ANY device depends on how much temperature rise it can withstand before catastrophic failure occurs. As
@cjdelphi noted, a block of ice with the resistor embedded in the ice will turn a 1/4 watt resistor into a 10 watt resistor at least until the ice melts. Even so, the heat generated must reach the surrounding environment and there is always
thermal resistance in the way. So, given two resistors of the same physical dimensions, one rated 1/4 watt and the other 1/2 watt (and labeled "mini") there are several options. The most obvious one is the label "mini" is meaningless as far as power dissipation is concerned. It could just means smaller than regular size. Then, along with the small size, come limitations on power dissipation that are probably comparable or identical to the resistor identified as 1/4 watt. In other words, both resistors, despite the labeling and hype, are both 1/4 watt resistors.
There is a second possibility. The 1/2 watt resistor in a 1/4 watt-sized package may be of high-temperature construction, designed to run hotter without failing, changing resistance, or other undesirable characteristics. A lot depends on material selection. But one thing is certain: the "mini" resistor will run a LOT hotter with 1/2 watt dissipation than the 1/4 watt resistor with 1/4 watt dissipation. This is simple thermodynamics. There is the same surface area from which to dissipate heat (usually by air convection) but twice the power to dissipate. Ergo, hotter resistor with 1/2 watt dissipation, no matter what it says on the package.
If I were you, I would do what
@dorke suggested: hook both resistors up to a power supply and apply the same quarter-watt of power to both. I am guessing, based on their sizes being virtually identical, that both will get equally warm. Repeat this test using the physically larger 1/2 watt resistor and you should feel a noticeable difference in temperature because the "real" 1/2 watt resistor has a larger surface area from which to dissipate the quarter watt of power.
If all the resistors are the same value, just connect them in series and apply enough voltage so you are supplying at least 1/4 watt of power to each one (set total power, equal to DC supply voltage times current, to 0.75 watts). Use your "temperature calibrated" thumb and forefinger to measure the temperature rise. Note you should try this only with small-valued resistors so the applied voltage is "safe" to work with. If you have a 5 V DC supply, the resistance should be about 33 ohms each for three in series.
