The Poynting vector is a vector that points in the direction of the flow of electromagnetic energy. It is a surface power density, measured in in SI units. It is defined as
where is the electric field and is the magnetic field. Even static fields can lead to a movement of energy.
Examples#
> The magnetic field is dependent on the distance $s$ from the center of the cylinder > $$\mathbf{B}(s)=\frac{\mu_{0}I}{2\pi s}\hat{\boldsymbol{\phi}},\qquad 0\leq s\leq a> Numerical value aside, notice the presence of the minus sign: it means that the vector is going towards the *inside* of the cylinder. In a way, energy is being "sucked in" by the center of the cylinder. This is what happens in, say, a simple electrical wire. > [!example]- Coaxial cylinders > A similar situation is with two cylinders, with one embedded inside of the other. The length is still $L$ for both, with small and large radii $a$ and $b$. The electric potential difference $V$ is now applied in between the cylinders, so the electric field goes out radially. The current is still passing through the center, so the magnetic field is the same when $a\leq s\leq b$. This is the standard form of a cylindrical [[capacitor]]. > > ![[Diagram Poynting vector coaxial cylinders.svg|80%]] > > The electric field can be found by integrating the potential using the definition from $a$ to $b$ (remember the minus sign in front of the integral!), which gives the line charge density, and then through the usual field for a line charge density. > $$V=-\int_{a}^{b}\mathbf{E}\cdot d\mathbf{s}=\frac{\lambda}{2\pi \varepsilon_{0}}\ln\left( \frac{b}{a} \right)\quad\Rightarrow \lambda=\frac{2\pi \varepsilon_{0}V}{\ln(b/a)}The Poynting vector therefore is
> The Poynting vector is > $$S=\frac{1}{\mu_{0}} \frac{\mu_{0}I}{2\pi s} \frac{V}{s\ln(b/a)}\hat{\mathbf{s}}\times \hat{\boldsymbol{\phi}}=\frac{VI}{2\pi s^{2}\ln(b/a)}\hat{\mathbf{z}}Assuming we're dealing with a large , we can use the field for an infinite straight wire
The energy now goes down the length of the cylinders.