Show that average value of radiant flux density S over a single period T is given by S= <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <mi mathvariant="bold-italic">C</mi> <msub> <mrow> <mrow> <mi mathvariant="bold-italic">μ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> <msubsup> <mrow> <mrow> <mi mathvariant="bold-italic">E</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msubsup> </math> .

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Electromagnetic Waves Physics Ncert Solutions Class 12th Physics Class 12th Physics NCERT Exemplar Solutions Class 12th Chapter Eight

Show that average value of radiant flux density S over a single period T is given by S= $\frac{1}{2 C μ_{0}} E_{0}^{2}$ .

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Answered by

Payal Gupta | Contributor-Level 10

4 months ago

This is a short answer type question as classified in NCERT Exemplar

As we know radiant flux density S= $\frac{1}{μ_{0}}$ (E $\times B$ )= c^{2
$?$}₀(E $\times B$ )

Suppose electromagnetic waves be propagating along x-axis. The electric field vector of

Electromagnetic wave be along y-axis and magnetic field vector be along z-axis. Therefore,

E=E₀cos (kx-wt)

B=B₀cos (kx-wt)

E $\times$ B= (E₀B₀)cos²(kx-wt)

S= c^{2
$ε_{0}$}(E $\times B$ )

Radiant flux density over a complete cycle is

S_av= c^{2
$?$}₀|E_{0
$\times$}B₀| $\frac{1}{T} \int_{0}^{T} {c o s}^{2} (k x - w t) d t$

= c^{2
$?$} ₀|E₀B₀| $\frac{1}{T} \times \frac{T}{2}$

= $\frac{C^{2}}{2} ε$ ₀E₀( $\frac{E_{0}}{c}$ )

= $\frac{c}{2} ε$ ₀E₀²= $\frac{c}{2} \times$ $\frac{1}{c^{2} μ_{0}}$ E₀²

= $\frac{E_{0}^{2}}{2 μ_{0} c}$

This is a short answer type question as classified in NCERT ExemplarAs we know radiant flux density S=  <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </math> (E <math> <mo>×</mo> <mi>B</mi> </math> )= c2 <math> <mi>?</mi> </math> 0(E <math> <mo>×</mo> <mi>B</mi> </math> )Suppose electromagnetic waves be propagating along x-axis. The electric field vector ofElectromagnetic wave be along y-axis and magnetic field vector be along z-axis. Therefore,E=E0cos (kx-wt)B=B0cos (kx-wt)E <math> <mo>×</mo> </math> B= (E0B0)cos2(kx-wt)S= c2 <math> <msub> <mrow> <mrow> <mi>ε</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </math> (E <math> <mo>×</mo> <mi>B</mi> </math> )Radiant flux density over a complete cycle isSav= c2 <math> <mi>?</mi> </math> 0|E0 <math> <mo>×</mo> </math> B0| <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> </mfrac> <msubsup> <mo>∫</mo> <mrow> <mn>0</mn> </mrow> <mrow> <mi>T</mi> </mrow> </msubsup> <mrow> <mrow> <msup> <mrow> <mrow> <mi>c</mi> <mi>o</mi> <mi>s</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <mo>(</mo> <mi>k</mi> <mi>x</mi> <mo>-</mo> <mi>w</mi> <mi>t</mi> <mo>)</mo> <mi>d</mi> <mi>t</mi> </mrow> </mrow> </math>      = c2 <math> <mi>?</mi> </math> 0|E0B0| <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> </mfrac> <mo>×</mo> <mfrac> <mrow> <mrow> <mi>T</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> </math>      = <math> <mfrac> <mrow> <mrow> <msup> <mrow> <mrow> <mi>C</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mi>ε</mi> </math> 0E0( <math> <mfrac> <mrow> <mrow> <msub> <mrow> <mrow> <mi>E</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> <mrow> <mrow> <mi>c</mi> </mrow> </mrow> </mfrac> </math> )     = <math> <mfrac> <mrow> <mrow> <mi>c</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mi>ε</mi> </math> 0E02= <math> <mfrac> <mrow> <mrow> <mi>c</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </mfrac> <mo>×</mo> </math> <math> <mfrac> <mrow> <mrow> <mn>1</mn> </mrow> </mrow> <mrow> <mrow> <msup> <mrow> <mrow> <mi>c</mi> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msup> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> </mrow> </mrow> </mfrac> </math>  E02     = <math> <mfrac> <mrow> <mrow> <msubsup> <mrow> <mrow> <mi>E</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> </mrow> </mrow> </msubsup> </mrow> </mrow> <mrow> <mrow> <mn>2</mn> <msub> <mrow> <mrow> <mi>μ</mi> </mrow> </mrow> <mrow> <mrow> <mn>0</mn> </mrow> </mrow> </msub> <mi>c</mi> </mrow> </mrow> </mfrac> </math>

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Show that average value of radiant flux density S over a single period T is given by S= $\frac{1}{2 C μ_{0}} E_{0}^{2}$ .

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Show that average value of radiant flux density S over a single period T is given by S= 1 2 C μ 0 E 0 2 .

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Show that average value of radiant flux density S over a single period T is given by S= $\frac{1}{2 C μ_{0}} E_{0}^{2}$ .