Home electrical stoves normally have four or five places for cooking. What is the name of each one of these places where a cooking pot is placed?
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It's called a stove burner or just a burner, at least in American English. Colin Fine notes that hob is more common in British English, so it may depend on where your expected audience is.
The use of "burner" dates back to when all stoves actually burned fuel like wood, coal, or gas, but the word still used for electric stoves that don't actually burn anything. Hob, on the other hand, originally meant a niche or shelf inside a fireplace where something could be kept warm.
Industry and retail terms include heating element or cooking element, or element if the context is clear. Technically, the heating element is any piece of equipment that supplies heat, so to distinguish the type in your photo from the heating elements in the oven or broiler, you might find it labeled as a cooktop element or surface element.
Many people, at least in the U.S., will call it a burner. Even people who do not have natural gas or propane stoves (which indeed burn their fuel) will be familiar with them, and understand the analogy. Element after all is somewhat clinical. AHD lists this as a definition of burner:
One that burns, especially:
a. A device, as in a furnace, stove, or gas lamp, that is lighted to produce a flame.
b. A device on a stovetop, such as a gas jet or electric element, that produces heat.
Others may refer to it as the coil (heating coil, cooking coil, etc.) referencing its shape, and particularly when drawing a distinction of type against, say, flat-top glass/ceramic heating or induction cooking elements.
And of course, you will find mashups like coil burner or burner element.
Colin Fine in a comment refers to hob as the British term, though I previously thought that it referred to the entire stovetop or cooktop as it is known in AmE. Hob is generaly unknown in AmE.