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I was trying to derive the formula $P = IV$, and I figured out to start from the basis that power is just work divided by time. In this case, I thought that I could use the formula $Vq = PE$ and substitute work by the equivalent expression of PE, so the formula becomes $P = Vq/t$, and $q/t$ is $I$, so the final formula is $P = VI$, which is right.

Does this mean that work and energy are the same thing? They have the same units, and it seems like at times they are equivalent, since I could substitute one for the other. If they are the same, why are they 2 different things?

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Work is energy that has been expended in the performance of a task. Energy is work which has been stored up for later use.

You can think of work and energy as money; energy is money in the bank while work is money being spent.

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Work is one specific form of energy. It is specific in that it flows between a system and its surroundings. Heat is the other form of energy that flows between a system and its surroundings.

Work comes in two forms, mechanical or other. Mechanical work is the expression $\delta w = -p_{ext} dV$ that appears in thermodynamic equations such as the first law of thermodynamics. Mechanical work appears when a system changes its volume. A system that expands does work, and a system that contracts has work done on it. The term "other work" is energy flow in forms as electricity or light. You are dealing with other work as electricity.

Energy is a content rather than a flow. External energy is in the form of kinetic energy $E_K$ due to motion of the system in a reference frame or potential energy $E_p$ due to position of a system in a force field (e.g. gravity). Internal energy $U$ is the inherent energy of the system due to its very existence.

The conservation of energy states that total energy $E_T = E_K + E_p + U$ of the universe does not change for any process. Energy may flow between a system and the surroundings as heat, mechanical work, or other work. Potential energy may convert to kinetic energy. In the end, the total energy content of the universe remains constant.

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Does this mean that work and energy are the same thing?

Work is a transfer of energy. So $W=\Delta E$. They are not the same thing, but they are closely related.

If it helps, energy and work have the same relationship as momentum and impulse.

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  • $\begingroup$ Is it fair to say that work the inaction of energy? $\endgroup$
    – Bored Comedy
    Commented Apr 7, 2021 at 3:25
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    $\begingroup$ It might be helpful to note that writing $W=\Delta E$ is a simplification that needs to be applied carefully. $\endgroup$
    – BioPhysicist
    Commented Apr 7, 2021 at 3:34
  • $\begingroup$ Sorry, I meant enaction instead of inaction. $\endgroup$
    – Bored Comedy
    Commented Apr 7, 2021 at 3:49
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Work is the transfer of energy, power is the rate that work is done and energy is the ability of something to do work.

Energy is typically only meaningful in changes, so work usually turns out to be a more physically useful quantity.

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Work is the transfer of energy by mechanical means, that is through force over a distance. If energy is the currency, work is one kind of financial transaction.

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  • $\begingroup$ Mechanical work is one type of work. Electrical work and magnetic work are other types. It might be more accurate to say that work is energy transfer not involving heat transfer or mass transfer—that is, energy transfer not driven by a temperature or chemical potential gradient, respectively. $\endgroup$
    – Chemomechanics
    Commented Apr 7, 2021 at 6:46

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