What is Electric Power?
In general physics terms, power is defined as the rate at which energy is transferred (or transformed).
So, first, what is energy and how is it transferred? It's hard to state simply, but energy is basically the ability of something to move something else. There are many forms of energy: mechanical, electrical, chemical, electromagnetic, thermal, and many others.
Energy can never be created or destroyed, only transferred to another form. A lot of what we're doing in electronics is converting different forms of energy to and from electric energy. Lighting LEDs turns electric energy into electromagnetic energy. Spinning motors turns electric energy into mechanical energy. Buzzing buzzers makes sound energy. Powering a circuit off a 9V alkaline battery turn chemical energy into electrical energy. All of these are forms of energy transfers.
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Electric energy in particular, begins as electric potential energy -- what we lovingly refer to as voltage. When electrons flow through that potential energy, it turns into electric energy. In most useful circuits, that electric energy transforms into some other form of energy. Electric power is measured by combining both how much electric energy is transferred, and how fast that transfer happens.
Producers and Consumers
Each component in a circuit either consumes or produces electric energy. A consumer transforms electric energy into another form. For example, when an LED lights up, electric energy is transformed into electromagnetic. In this case, the lightbulb consumes power. Electric power is produced when energy is transferred to electric from some other form. A battery supplying power to a circuit is an example of a power producer.