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zenterix

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- Homework Statement
- While reading a chapter about Poynting vector I reached a passage with what seems like a typo.

- Relevant Equations
- If it is a typo, I am still not sure how to fix it.

If it is not, then I am not sure what exactly was done in the algebra.

Here is a chapter from MIT OCW's 8.02 Electromagnetism course.

At the end of page 14 is section 13.6 "Poynting Vector". The calculations I am interested in are on page 15.

There is a passage that seems to have a typo in it. Let me try to show why despite recognizing a typo I am unsure of what the correct version would be.

Consider a plane wave passing through the infinitesimal volume element below

The total energy in the EM fields in the volume element is

$$dU=UAdx=(U_E+U_B)Adx=\left (\frac{1}{2}\epsilon_0E^2+\frac{1}{2\mu_0}B^2\right )Adx\tag{1}$$

$$=\frac{1}{2}\left (\epsilon_0E^2+\frac{B^2}{\mu_0}\right )Adx\tag{2}$$

The rate of change of energy per unit area is

$$S=\frac{dU}{dt}\frac{1}{A}=\frac{c}{2}\left (\epsilon_0E^2+\frac{B^2}{\mu_0}\right )\tag{2a}$$

where I have used the fact that the EM wave is traveling with speed ##c## and so ##dx=cdt##.

It can be shown that ##\frac{E}{B}=c=\frac{1}{\sqrt{\epsilon_0mu_0}}##, the speed of light.

The chapter then rewrites (2) but the expression seems to contain a typo. Here is the exact passage as it appears in the chapter

$$S=\frac{1}{2}\left (\epsilon_0E^2+\frac{B^2}{\mu_0}\right )=\frac{cB^2}{\mu_0}=c\epsilon_0E^2=\frac{EB}{\mu_0}\tag{3}$$

When I rewrite (2) I get

$$S=\frac{1}{2}\left (c\epsilon_0E^2+c\frac{B^2}{\mu_0}\right )$$

$$=\frac{1}{2}\left (\epsilon_0\frac{E^3}{B}+\frac{EB}{\mu_0}\right )$$

What am I missing?

At the end of page 14 is section 13.6 "Poynting Vector". The calculations I am interested in are on page 15.

There is a passage that seems to have a typo in it. Let me try to show why despite recognizing a typo I am unsure of what the correct version would be.

Consider a plane wave passing through the infinitesimal volume element below

The total energy in the EM fields in the volume element is

$$dU=UAdx=(U_E+U_B)Adx=\left (\frac{1}{2}\epsilon_0E^2+\frac{1}{2\mu_0}B^2\right )Adx\tag{1}$$

$$=\frac{1}{2}\left (\epsilon_0E^2+\frac{B^2}{\mu_0}\right )Adx\tag{2}$$

The rate of change of energy per unit area is

$$S=\frac{dU}{dt}\frac{1}{A}=\frac{c}{2}\left (\epsilon_0E^2+\frac{B^2}{\mu_0}\right )\tag{2a}$$

where I have used the fact that the EM wave is traveling with speed ##c## and so ##dx=cdt##.

It can be shown that ##\frac{E}{B}=c=\frac{1}{\sqrt{\epsilon_0mu_0}}##, the speed of light.

The chapter then rewrites (2) but the expression seems to contain a typo. Here is the exact passage as it appears in the chapter

$$S=\frac{1}{2}\left (\epsilon_0E^2+\frac{B^2}{\mu_0}\right )=\frac{cB^2}{\mu_0}=c\epsilon_0E^2=\frac{EB}{\mu_0}\tag{3}$$

When I rewrite (2) I get

$$S=\frac{1}{2}\left (c\epsilon_0E^2+c\frac{B^2}{\mu_0}\right )$$

$$=\frac{1}{2}\left (\epsilon_0\frac{E^3}{B}+\frac{EB}{\mu_0}\right )$$

What am I missing?

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