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Poynting Vector | What is Poynting Vector | Poynting Vector Formula

Poynting Vector | What is Poynting Vector | Poynting Vector Formula :- The Poynting vector is a physical quantity describing the magnitude and direction of energy flow in electromagnetic waves. The Poynting vector expresses the energy or energy flux passing through a unit area per second.

Poynting Vector $\displaystyle \overrightarrow{S}$ can be represented by the cross product $\displaystyle \overrightarrow{S}=\frac{1}{\mu }\left( \overrightarrow{E}\text{ }\times \text { }\overrightarrow{B} \right)$ , where μ is the permeability of the medium through which the radiation passes, E is the electric field intensity and B is the magnetic field intensity.

The direction of the vector product $\displaystyle \overrightarrow{E}\text{ }\times \text{ }\overrightarrow{B}$ is perpendicular to the plane of the vectors E and B and it is the direction of wave propagation. The Poynting vector describes the power crossing per unit area, so its units are watt/meter².

Poynting Vector | What is Poynting Vector | Poynting Vector Formula

Proof

If dU energy passes through dA area perpendicular to it in dt time, then the magnitude of Poynting vector is given by

$\displaystyle S=\frac{dU}{dAdt}\text{ }.....(1)$

In the above figure the total energy stored in volume element dV,

$\displaystyle dU\text{ }=\text{ }udV\text{ }.....(2)$

here u (energy density) = u_E + u_B

$\displaystyle \Rightarrow u={{u}_{E}}+{{u}_{B}}=\frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}+\frac{1}{2}\frac{{{B}^{2}}}{{{\mu }_{0}}}$

Putting E = cB and B = E/c in the above expression

$\displaystyle u=\frac{1}{2}{{\varepsilon }_{0}}E(cB)+\frac{1}{2}\frac{B}{{{\mu }_{0}}}\frac{E}{c}$

Now as $\displaystyle {{c}^{2}}=\frac{1}{{{\mu }_{0}}{{\varepsilon }_{0}}}$ , therefore $\displaystyle {{\varepsilon }_{0}}c=\frac{1}{{{\mu }_{0}}c}$ . now putting the value of ε₀c we get,