Confusion about the Energy Density of EM waves

[PumpkinCougar95]

I am a bit confused about the energy density in an EM wave. why do we take the Peak value of E vector while calculating the energy density?

Like if the E field is $E_0 Sin(kx-wt)$ what is the energy density of the EM wave(Magnetic + Electric)?

is it A) $\frac {e_0E_0^2}{2}$ or B) $e_0E_0^2$ ?

in the A) part I have used the Average value of E field as $<E^2> ~ = \frac {E_0^2 }{2}$

 

[jtbell]

I am a bit confused about the energy density in an EM wave.

I have used the Average value of E field as $<E^2> ~ = \frac {E_0^2 }{2}$

Is the electric field the only field in an electromagnetic wave?

 

[PumpkinCougar95]

No, There is the B field too.

so the total Energy density is:
$$u = \frac {\epsilon_0E^2}{2} + \frac {B^2}{2 \mu_0}$$
My question is do we take $E=E_0$ and $B=B_0$ while calculating? Or do we Use $E= \frac {E_0}{ sqrt{2} }$ and $B=\frac {B_0}{sqrt{2}}$ ? Because in multiple books i have seen $E=E_0$ and $B=B_0$ which doesn't make any sense to me.

 

[jtbell]

Are you looking for the "instantaneous" energy density at a point in space and time in the wave, or the average energy density over a volume of space or period of time? Your equation for $u$ is the "instantaneous" energy density at any point in the wave, using the values of $E$ and $B$ at that point. Setting $E = E_0$ and $B = B_0$ gives you the maximum energy density, $u_0$, at the maxima of the waves. Setting $E = E_0 / \sqrt 2$ and $B = B_0 / \sqrt 2$ gives you the average energy density, $\langle u \rangle$, for a sinusoidal wave.

More carefully, $$\langle u \rangle = \frac 1 2 \varepsilon_0 \langle E^2 \rangle + \frac 1 {2 \mu_0} \langle B^2 \rangle = \frac 1 4 \varepsilon_0 E_0^2 + \frac 1 {4 \mu_0} B_0^2$$

Most books take one more step, using the relationship between $E$ and $B$ in an electromagnetic wave. I'll let you fill in that step!

Different books, or even different points in the same book, may be talking about different kinds of energy density (instantaneous versus average), and sometimes you have to read carefully to see which is which. The mathematical notation may not always be consistent, unfortunately.

 

[PumpkinCougar95]

Oh Ok. Thanks for your help!