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The following definition of power is given in Engineering Circuit Analysis by Hayt, Kemmerly and Durbin, 8th edition Power Definition Also, as they say that they have already defined power, their earlier definition of power is:

Power is defined as the rate at which work is done or energy is expended.

My doubt is what is the use of one coulomb of charge in the definition of power? Whether we pass 5 coulomb of charge or 10 coulomb of charge, as long as 1 joule of energy is being expended in 1 second, the power will be 1 watt.

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You are quite right. Including 1 coulomb in the definition of power is thoroughly confusing – and confused. [It limits the scope of the definition to the case when the potential difference is 1 volt !]

The second definition is fine, though I'd change 'expended' to the more general 'transferred'.

You'll agree that the second definition isn't specifically electrical. Neither should it be. The concept of power can be applied where there is any sort of energy transfer, so it might be mechanical to mechanical, mechanical to thermal, electrical to thermal... Although the famous formula Power$=VI$ involves electrical quantities, it isn't a different definition of power that we have to use in electrical work; the formula is easily derived from the general definition of power that you have quoted (the second definition).

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