Ncert Exercises & Solutions

A plane EM wave travelling along z-direction is described by $\vec{E} = E_{0} \sin (kz - ωt) \hat{i} and \vec{B} = B_{0} \sin (kz - ωt) \hat{j} . Show that :$

(i) the average energy density of the wave is given by $U_{av}=\frac14\varepsilon_0E^2_0+\frac14\frac{B^2_0}{\mu_0}$

(ii) the time averaged intensity of the wave is given by $I_{av}=\frac12c\varepsilon_0E^2_0$ .

Hint: The average energy density of the wave is distributed equally between the electric field and the magnetic field.

(I) Step 1: The electromagnetic wave carries energy which is due to electric field vector and magnetic field vector. In an electromagnetic wave, E and B vary from point to point and from moment to moment. Let E and B be their time averages.

$The energy density due to electric field E is$

$U_{E} = \frac{1}{2} ε_{0} E^{2}$

$The energy density due to magnetic field B is :$

$U_{B} = \frac{1 B^{2}}{2 μ_{0}}$

$Total average energy density of electromagnetic wave :$

$U_{av} = U_{E} = \frac{1}{2} ε_{0} E^{2} + \frac{1}{2} \frac{B^{2}}{μ_{0}}$

$Step 2 : Let the EM wave be propagating along z - direction . The electric field vector and magnetic field vector be represented by$

$E = E_{0} \sin (kz - ωt)$

$B = B_{0} \sin (kz - ωt)$

$The time average value of E^{2} over complete cycle = \frac{E_{0}^{2}}{2}$

$The time average value of B^{2} over complete cycle = \frac{B_{0}^{2}}{2}$

$U_{av} = \frac{1}{2} \frac{ε_{0} E_{0}^{2}}{2} + \frac{1}{2} μ_{0} (\frac{B_{0}^{2}}{2})$

$= \frac{1}{4} ε_{0} E_{0}^{2} + \frac{B_{0}^{2}}{4 μ_{0}}$

$(ii) Step 3 : We know that E_{0} = {cB}_{0} and c = \frac{1}{\sqrt{μ_{0} ε_{0}}}$

$∴ \frac{1}{4} \frac{B_{0}^{2}}{μ_{0}} = \frac{1}{4} \frac{E_{0}^{2} / c^{2}}{μ_{0}} = \frac{E_{0}^{2}}{4 μ_{0}} \times μ_{0} ε_{0} = \frac{1}{4} ε_{0} E_{0}^{2}$

$∴ U_{B} = U_{E}$

$Hence, U_{av} = \frac{1}{4} ε_{0} E_{0}^{2} + \frac{1}{4} \frac{B_{0}^{2}}{4 μ_{0}}$

$= \frac{1}{4} ε_{0} E_{0}^{2} + \frac{1}{4} ε_{0} E_{0}^{2}$

$= \frac{1}{2} ε_{0} E_{0}^{2} = \frac{1}{2} \frac{B_{0}^{2}}{μ_{0}}$

$Time average intensity of the wave :$

$I_{av} = U_{av} C = \frac{1}{2} ε_{0} E_{0}^{2} c = \frac{1}{2} ε_{0} E_{0}^{2}$

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