JEE Main & Advanced Physics Wave Optics / तरंग प्रकाशिकी Properties of EM Waves

(1) Speed : In free space it's speed

\[c=\frac{1}{\sqrt{{{\mu }_{0}}{{\varepsilon }_{0}}}}=\frac{{{E}_{0}}}{{{B}_{0}}}=3\times {{10}^{8}}m/s.\]

In medium \[v=\frac{1}{\sqrt{\mu \varepsilon }}\]; where \[{{\mu }_{0}}=\] Absolute permeability, \[{{\varepsilon }_{0}}=\] Absolute permittivity.

(2) Energy : The energy in an EM waves is divided equally between the electric and magnetic fields.

Energy density of electric field \[{{u}_{e}}=\frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}\], Energy density of magnetic field \[{{u}_{B}}=\frac{1}{2}\frac{{{B}^{2}}}{{{\mu }_{0}}}\]

The total energy per unit volume is \[u={{u}_{e}}+{{u}_{m}}\]\[=\frac{1}{2}{{\varepsilon }_{0}}{{E}^{2}}+\frac{1}{2}\frac{{{B}^{2}}}{{{\mu }_{0}}}\]. Also  \[{{u}_{av}}=\frac{1}{2}{{\varepsilon }_{0}}E_{0}^{2}\]\[=\frac{B_{0}^{2}}{2{{\mu }_{0}}}\]

(3) Intensity (I) : The energy crossing per unit area per unit time, perpendicular to the direction of propagation of EM wave is called intensity.

i.e. \[I=\frac{\text{Total EM energy}}{\text{Surface area }\times \text{Time}}=\frac{\text{Total energy density}\times \text{Volume}}{\text{Surface area }\times \text{ Time}}\] \[\Rightarrow \]\[I={{u}_{av}}\times c=\frac{1}{2}{{\varepsilon }_{0}}E_{0}^{2}c=\frac{1}{2}\frac{B_{0}^{2}}{{{\mu }_{0}}}.c\frac{Watt}{{{m}^{2}}}.\]

(4) Momentum : EM waves also carries momentum, if a portion of EM wave of energy u propagating with speed c, then linear momentum \[=\frac{\text{Energy (}u\text{)}}{\text{Speed (}c\text{)}}\]

If wave incident on a completely absorbing surface then momentum delivered \[p=\frac{u}{c}\]. If wave incident on a totally reflecting surface then momentum delivered \[-p=\frac{2u}{c}\].

(5) Poynting vector\[(\vec{S}).\] : In EM waves, the rate of flow of energy crossing a unit area is described by the Poynting vector.

(i) It's unit is \[Watt/{{m}^{2}}\] and \[\vec{S}=\frac{1}{{{\mu }_{o}}}(\vec{E}\times \vec{B})={{c}^{2}}{{\varepsilon }_{0}}(\overrightarrow{E}\times \overrightarrow{B})\].

(ii) Because in EM waves \[\overrightarrow{E}\] and \[\overrightarrow{B}\] are perpendicular to each other, the magnitude of \[\overrightarrow{S}\] is \[\,|\vec{S}|\,=\frac{1}{{{\mu }_{0}}}E\,B\,\sin {{90}^{o}}=\frac{EB}{{{\mu }_{0}}}=\frac{{{E}^{2}}}{\mu \,C}\].

(iii) The direction of \[\overrightarrow{S}\] does not oscillate but it's magnitude varies between zero and a maximum \[\left( {{S}_{\max }}=\frac{{{E}_{0}}{{B}_{0}}}{{{\mu }_{0}}} \right)\] each quarter of a period.

(iv) Average value of poynting vector is given by \[\overline{S}=\frac{1}{2{{\mu }_{0}}}{{E}_{0}}{{B}_{0}}=\frac{1}{2}{{\varepsilon }_{0}}E_{0}^{2}c=\frac{cB_{0}^{2}}{2{{\mu }_{0}}}\]

The direction of the poynting vector \[\overrightarrow{S}\] at any point gives the wave's direction of travel and direction of energy transport the point.

(6) Radiation pressure : Is the momentum imparted per second pre unit area. On which the light falls.

For a perfectly reflecting surface \[{{P}_{r}}=\frac{2S}{c}\]; S = Poynting vector; c = Speed of light

For a perfectly absorbing surface \[{{P}_{a}}=\frac{S}{c}.\]

(7) Wave impedance (Z) : The medium offers hindrance to the propagation of wave. Such hindrance is called wave impedance and it is given by \[Z=\sqrt{\frac{\mu }{\varepsilon }}=\sqrt{\frac{{{\mu }_{r}}}{{{\varepsilon }_{r}}}}\sqrt{\frac{{{\mu }_{0}}}{{{\varepsilon }_{0}}}}\]

For vacuum or free space \[Z=\sqrt{\frac{{{\mu }_{0}}}{{{\varepsilon }_{0}}}}=376.6\,\Omega .\]