We all know that poynting vector points in the direction of the wave propagation and it's value of power supplied to a unit area.

So while deriving poynting theorm, the energy flown out of the system can be written as

$$\int S \cdot da$$ Where da is an infinitesimally small area element.

But I don't understand why do we use $S$ and $da$ as vector i.e. why do we write energy flown out of the system as

$$\int \vec S \cdot d\vec a$$


1 Answer 1


The Poynting vector does not always point in the direction of wave propagation. In vacuum it does, but in a general medium the electric field has a component in the direction of propagation, while $\vec E \times \vec B$ is clearly perpendicular to $\vec E$.

It's also worth noting that the Poynting vector is defined even for static fields, so there need not be any notion of waves at all.

But to address your actual question, the flux of a vector field through a surface depends on the relative orientation of the surface with respect to the field. You need the notion of a dot product to formalize this.

Presumably if you're talking about Poynting's theorem you've already learned about Gauss' law, so you've seen all of this before.


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