The average value of electric energy density | vedclass

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Question

(B) The instantaneous electric energy density is given by $u_E = \frac{1}{2} \varepsilon_0 E^2$. For an electromagnetic wave,the electric field varies

Vedclass · Accepted Answer

$\frac{1}{4}\varepsilon_0 E_0^2$

Vedclass · Answer

(B) The instantaneous electric energy density is given by $u_E = \frac{1}{2} \varepsilon_0 E^2$. For an electromagnetic wave,the electric field varies sinusoidally as $E = E_0 \sin(kx - \omega t)$. The average value of $\sin^2(kx - \omega t)$ over a complete cycle is $\frac{1}{2}$. Therefore,the average electric energy density is $\langle u_E \rangle = \frac{1}{2} \varepsilon_0 \langle E^2 \rangle = \frac{1}{2} \varepsilon_0 (E_0^2 \cdot \frac{1}{2}) = \frac{1}{4} \varepsilon_0 E_0^2$.

The average value of electric energy density in an electromagnetic wave is:

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