# Why is power considered to be a scalar quantity?

I have always been a bit confused on deciding on scalar and vector quantities. Most of the time, my intuition gives me opposite to the right answer. So now I desperately want to know why power including work is a scalar quantity.

• by definition? – AccidentalFourierTransform Oct 4 '17 at 14:58
• well, yeah. When I asked my Teacher about this, he talked about stuff like dot product etc. But I wanna know exactly what it is. – bur Oct 4 '17 at 15:02
• Power is not always a scalar quantity. For example, with Electrodynamics, the Poynting Theorem states that power has direction showing the flow of power in an EM field. Actually, this is power density showing the power flow per area. – K7PEH Oct 4 '17 at 15:04
• Oh , ohkay, this is the exam question I had to answer: – bur Oct 4 '17 at 15:07
• (c) The following list contains scalar and vector quantities. Underline all the scalar quantities. acceleration , force , mass , power , temperature , weight  – bur Oct 4 '17 at 15:08

Power on its own is defined as being the rate of energy transfer, and it has no additional information as to its direction, so it is a scalar. However, there is a vectorial quantity which is related to power, known as the Poynting vector.

Given say, an electric field $\mathbf{E}$ and magnetic field $\mathbf B$, the Poynting vector is defined as,

$$\mathbf S = \frac{1}{\mu_0}\mathbf E \times \mathbf B$$

which is the power in the direction of $\mathbf S$, per unit area. Thus, if we want to know the power going through a surface $A$, it would be,

$$P = \iint_A \mathbf S \, \cdot \mathrm d\mathbf A.$$

Thus, power on its own is a scalar quantity, but we do have a notion of direction for power which is encoded in the Poynting vector, or analogues of it for other phenomena.

• okay, alright. So in whichever form it is, Power in itself ends up being a scalar? – bur Oct 4 '17 at 15:22
• So then, what about work as a scalar quantity?. It's got Force (vector) and distance. How then can I exactly say it's a scalar quantity. – bur Oct 4 '17 at 15:30
• @bur Yes, power is a scalar, but it may encoded in quantities which are not scalar, though this can be said of many things. – JamalS Oct 4 '17 at 15:32

Whether a quantity is a scalar or vector or higher-ranked tensor actually depends on how they are used to model a physical process and how they need to transform under coordinate transformations.

Vectors have certain transformation properties, most notably rotational, which are different from scalars, and tensors have transformation properties which are different from vectors, etc.

When introducing physics to beginners, the ideas of vectors and scalars are simplified, and they seem like arbitrary assignments to the students. At higher levels of physics, the concepts of rotation are brought in and are used to explain why a velocity is a vector, but mass is not, and so on. At even higher levels, tensors are introduced, and in relativity the electromagnetic field, previously modeled as a couple of vectors, $\vec{E}$ and $\vec{B}$, is presented in the form of a tensor, again due to transformation properties using the tools of mathematics.

Another conceptual construct is the 4-vector which has certain attractive transformation properties for modeling physical processes and doing calculations.

Power and work are some of those modeled, conceptual, important quantities which, in lower level physics can be treated as scalars.

• So in short, now I just have to simply know which is scalar and vector and wait till college to get things interesting and meaningful. – bur Oct 4 '17 at 15:42
• Probably so, unless your instructor wants to sit down with you and show you some of the specifics. – Bill N Oct 4 '17 at 15:45
• @AccidentalFourierTransform Thanks for the spelling corrections. Scalar doesn't flow off my fingers as readily as scaler. But who's counting? ;) – Bill N Oct 4 '17 at 15:46
• @BillN No problem :-) for a minute there, I was wondering whether that spelling was an AmE vs. BrE thing or something, as in "modeling"->"modelling" :-P – AccidentalFourierTransform Oct 4 '17 at 15:50

If you are talking about an intuitive way of understanding it you could think about it this way :

Power is defined as rate of change of work done.It is basically a measure of how quickly work is done. Now work is a scalar thus power doesn't have any direction associated with it.

• Alright, yes that's true – bur Oct 4 '17 at 15:49
• Why doesn't it have a direction if I'm doing work on a specific object and not on anything around it? I could "intuitively" say that I'm directing power toward the object. This doesn't satisfy me. If force is a vector, then why isn't the work done by the force a vector? It must come back to something bigger than intuition. – Bill N Oct 4 '17 at 15:49
• hmmm, basically I shouldn't jump to conclusions at this level. – bur Oct 4 '17 at 15:50
• @BillN Well, like you said, you are doing work on an object. But work itself is a scalar defined as the dot product of force and displacement. – rattle99 Oct 4 '17 at 15:53
• You know what, I really just need to by heart stuff right now. – bur Oct 4 '17 at 15:55

Actually you can write down a thrust four-vector as a rate of change of momentum in the proper frame. The time-component of this vector is power.