Poynting Vector | What is Poynting Vector | Poynting Vector Formula

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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 \vec{S}=\frac{1}{{{\mu }_{0}}}\left( \vec{E}\text{ }\times \text{ }\vec{B} \right), where μ0 is the permeability of free space, 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.

As we know that electric field (E) and magnetic field (B) vary sinusoidally [ \displaystyle E={{E}_{0}}\sin (kx-\omega t)\text{ }and\text{ }B={{B}_{0}}\sin (kx-\omega t) ] , hence the value of Poynting Vector varies with time.

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)

Poynting Vector

 

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

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

here u (energy density) = uE + uB

\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,

\displaystyle u=\frac{1}{2}\frac{EB}{{{\mu }_{0}}c}+\frac{1}{2}\frac{EB}{{{\mu }_{0}}c}

\displaystyle \Rightarrow u=\frac{EB}{{{\mu }_{0}}c}\text{ }.....(3)

By substituting the value of equation (3) in equation (2),

\displaystyle dU\text{ }=udV=\frac{EB}{{{\mu }_{0}}c}dV=\frac{EB}{{{\mu }_{0}}c}\times dAcdt=\frac{EB}{{{\mu }_{0}}}dAdt

Dividing both sides by dAdt we get,

\displaystyle \frac{dU}{dAdt}=\frac{EB}{{{\mu }_{0}}}

Hence from equation (1),

\displaystyle S=\frac{EB}{{{\mu }_{0}}}

In vector form,

\displaystyle \overrightarrow{S}=\frac{\overrightarrow{E}\times \overrightarrow{B}}{{{\mu }_{0}}}

 

 

 

Class 12th NCERT
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