why are some electrical plugs the same width on both prongs or have three instead of two prongs?

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Prongs might not be the right word for it.

In: Engineering
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That is to do with polarity. In America, the ground of a plug is the left side, the earth the middle, and the hot the right. When something has a metal casing, when it is plugged in it has the possibility to become charged if there is a surge in electricity. When this happens, normally a lightening strike nearby but even some older amps and TVs that don’t have the earth prong can become charged. A little shock will happen with the latter two, but if the lines are actually struck by lightening there would be a huge spike in charge inside the appliance. When the resistance of the ground wire is too high at that voltage and amperage, the electricity will travel through the earth prong into the breaker box, tripping the breakers, and dissipating the charge into the ground.

There are three connections. Hot, Neutral, and ground.

Under normal operation, hot and neutral carry the current, and neutral is at the same voltage as ground.

If you have a double insulated device(for example, with a plastic case), a single failure won’t cause part of the device you can touch to become electrified(so you don’t need a ground). Though if part of it can be accessed more easily, that will be connected to neutral. This is why a desk lamp will have a polarized plug(two prongs with one side bigger), it’s easier to touch the screw part of the bulb, so it’s supposed to be connected to neutral rather than hot.

As for ground, it’s supposed to be connected to stuff like metal housings, so that if a hot wire comes loose and touches the case, it will short circuit and trip the breaker or fuse instead of sitting there at high voltage waiting for someone to touch it.

The reason neutral isn’t used for this is that if the neutral line comes loose somewhere between the device and the breaker box, turning something on will cause neutral to become hot.

The third “prong” is ground. It is for safety of appliance and people when there’s too much electricity.
Phone charger doesn’t need it because it uses just a little bit of electricity. But washing machine needs big amount of power to operate so it must be grounded.
Of course it’s more complicated, but this gets the idea.