PX284 - R2 - poynting vector

the continuity equation:

- W = \frac{\partial u}{\partial t} + \vec{\nabla} \cdot \vec{S}

where, $u$ is the energy density, and $\vec{S}$ is the energy flux, $W$ is the rate of doing work per unit volume on free charges

from newton's third law:

- W = - \vec{E} \cdot ρ_{f} \vec{v} = - \vec{E} \cdot {\vec{J}}_{f} = - \vec{E} \cdot (\vec{\nabla} \times \vec{H} - \frac{\partial \vec{D}}{\partial t})

\vec{\nabla} \cdot (\vec{E} \times \vec{H}) = \vec{H} \cdot (\vec{\nabla} \times \vec{E}) - \vec{E} \cdot (\vec{\nabla} \times \vec{H})

\begin{aligned} - W & = \vec{\nabla} \cdot (\vec{E} \times \vec{H}) - \vec{H} \cdot (\vec{\nabla} \times \vec{E}) - \vec{E} \cdot \frac{\partial \vec{D}}{\partial t} \\ = \vec{\nabla} \cdot (\vec{E} \times \vec{H}) + \frac{\partial}{\partial t} (\frac{\vec{H} \cdot \vec{B} + \vec{D} \cdot \vec{E}}{2}) \end{aligned}

the poynting vector:

\vec{S} = \vec{E} \times \vec{H}

the energy density:

u = \frac{\vec{H} \cdot \vec{B} + \vec{D} \cdot \vec{E}}{2}

$\vec{S} ⊥ \vec{E}, \vec{H}$ and therefore $\vec{S} ∥ \vec{k}$
the magnitude:

S = E H = \frac{E^{2}}{Z} = Z H^{2}

$Z$ is frequency dependent, therefore, so is $S$

\begin{matrix} \vec{S} = \vec{E} \times \vec{H} ⟹ R e (\vec{E}) \times R e (\vec{H}) \\ | \vec{S} | = E_{0} \cos ω t H_{0} \cos ω t = \frac{E_{0}^{2}}{Z} \cos^{2} ω t \end{matrix}

average over time:

\bar{S} = \frac{E_{0}^{2}}{Z} \frac{1}{2} = \frac{E_{R M S}^{2}}{Z}

this is useful when considering waves of different frequencies, eg: sunlight: ${\bar{S}}_{e a r t h} = 1300 {Wm}^{- 2}$
the impedance of air $≃ Z_{0} = 377 Ω$
so, $E_{R M S} = \sqrt{Z_{0} {\bar{S}}_{e a r t h}} ≃ 700 {Vm}^{- 1}$
energy flux implies a momentum flux / radiation pressure: $E = p c$
the momentum flux, ie. a pressure:

\frac{\vec{S}}{c} = \frac{\vec{E} \times \vec{H}}{c}

eg: sunlight on earth: $| p_{r a d} | ≃ 4.3 \times 10^{- 6}$ Nm $^{- 2}$
radiation pressure can be significant
- large area, eg: solar sails
- more luminous radiation, eg: in very massive stars, the effect of $p_{r a d}$ can balance gravity, limiting the size and power of star

in a current carrying conductor

consider a cylindrical with a cross-sectional area, $A$ , carrying a current in the same direction as $\vec{E}$
from the right hand rule, $\vec{H}$ loops around the wire
$\vec{S} = \vec{E} \times \vec{H}$ , the pointing vector always points into the wire
$| \vec{E} | = V / L$ , $| \vec{H} | = I / 2 π r$ , so, the energy that enters the wire:

| \vec{S} | A = \frac{V I L 2 π r}{L 2 π r} = V I = E

this is equal to the energy dissipated in the wire