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1) \(\displaystyle \vec{B} = \vec{B_0} ~ sin( 2 \pi f t)\). This is not true in general and I have no idea what you are using for \(\displaystyle \vec{E}\). Are you trying to use an electromagnetic wave?

2) \(\displaystyle \vec{ \nabla } \cdot (B^2) = \vec{ \nabla } \cdot ( \vec{B} \cdot \vec{B} )\). This is meaningless. \(\displaystyle \vec{B} \cdot \vec{B}\) is a scalar quantity and \(\displaystyle \vec{ \nabla } \) can't be dotted with a scalar.

-Dan

Addendum: I just had a thought. Is the equation in 2) supposed to be \(\displaystyle \vec{ \nabla } ( B^2 )\) ?

1) Yes I tried to use an EM wave. Why is it wrong?

1) \(\displaystyle \vec{B} = \vec{B_0} ~ sin( 2 \pi f t)\). This is not true in general and I have no idea what you are using for \(\displaystyle \vec{E}\). Are you trying to use an electromagnetic wave?

2) \(\displaystyle \vec{ \nabla } \cdot (B^2) = \vec{ \nabla } \cdot ( \vec{B} \cdot \vec{B} )\). This is meaningless. \(\displaystyle \vec{B} \cdot \vec{B}\) is a scalar quantity and \(\displaystyle \vec{ \nabla } \) can't be dotted with a scalar.

-Dan

Addendum: I just had a thought. Is the equation in 2) supposed to be \(\displaystyle \vec{ \nabla } ( B^2 )\) ?

2) \(\displaystyle B^2 \) is a vector since it is given from \(\displaystyle \vec{B} \times \vec{E} \). Its direction is along the propagation of the EM wave. If \(\displaystyle \vec{E} \) is along y-axis and \(\displaystyle \vec{B} \) is along z-axis then \(\displaystyle \vec{B} \times \vec{E} \) would be along x-axis. I guess it vould be denoted as \(\displaystyle cB^2 \textbf{i} \)

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There's nothing wrong to use an EM wave. You simply didn't say that that's what you were doing. No worries.1) Yes I tried to use an EM wave. Why is it wrong?

2) \(\displaystyle B^2 \) is a vector since it is given from \(\displaystyle \vec{B} \times \vec{E} \). Its direction is along the propagation of the EM wave. If \(\displaystyle \vec{E} \) is along y-axis and \(\displaystyle \vec{B} \) is along z-axis then \(\displaystyle \vec{B} \times \vec{E} \) would be along x-axis. I guess it vould be denoted as \(\displaystyle cB^2 \textbf{i} \)

Also, standard notation is that \(\displaystyle B^2 \equiv \vec{B} \cdot \vec{B}\). The dot product gives a scalar as a result, not a vector. So in your notation you are saying that \(\displaystyle B^2\) is related to \(\displaystyle \vec{B} \times \vec{E}\). What, exactly, are you using for the definition? (Just as an afterthought. If E is in the +y direction and B is in the +z direction then \(\displaystyle \vec{B} \times \vec{E}\) is in the -x direction. As the Poynting vector gives the direction of the propagation of the wave as \(\displaystyle \vec{E} \times \vec{B}\) most people use that convention, which would be the +x direction in this case. But if you like \(\displaystyle \vec{B} \times \vec{E}\) we can work with that.)

Putting all that aside and saying that your equations up to this part are correct, most of the rest seems to be fine. My only remaining question (aside from those above) is how did you get \(\displaystyle V \approx 10^{-30}\)? (Units?) You didn't define a specific volume to be using and if you are using V to represent the "volume" of an electron your V is rather mystifying... The electron is considered to be a point particle and thus has no volume. Your number, however, seems to be correct given all of the above. So I guess \(\displaystyle B_0\) is surprisingly large. Perhaps that can be explained with the V you chose.

-Dan

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Well I used \(\displaystyle V \approx 10^{-30}\) because I found an approximation of the electron volume size to be that online. Thanks!There's nothing wrong to use an EM wave. You simply didn't say that that's what you were doing. No worries.

Also, standard notation is that \(\displaystyle B^2 \equiv \vec{B} \cdot \vec{B}\). The dot product gives a scalar as a result, not a vector. So in your notation you are saying that \(\displaystyle B^2\) is related to \(\displaystyle \vec{B} \times \vec{E}\). What, exactly, are you using for the definition? (Just as an afterthought. If E is in the +y direction and B is in the +z direction then \(\displaystyle \vec{B} \times \vec{E}\) is in the -x direction. As the Poynting vector gives the direction of the propagation of the wave as \(\displaystyle \vec{E} \times \vec{B}\) most people use that convention, which would be the +x direction in this case. But if you like \(\displaystyle \vec{B} \times \vec{E}\) we can work with that.)

Putting all that aside and saying that your equations up to this part are correct, most of the rest seems to be fine. My only remaining question (aside from those above) is how did you get \(\displaystyle V \approx 10^{-30}\)? (Units?) You didn't define a specific volume to be using and if you are using V to represent the "volume" of an electron your V is rather mystifying... The electron is considered to be a point particle and thus has no volume. Your number, however, seems to be correct given all of the above. So I guess \(\displaystyle B_0\) is surprisingly large. Perhaps that can be explained with the V you chose.

-Dan

I have played a bit more and ended up with this:

The volume obtained when using the classical electron radius is very close to the volume of the 1s orbital. Any comments about problems or further guidance is greately appreciated? In this derivation the charge density is assumed constant in the energy formulain the first post of this thread. But in 1s the charge density is not constant. Anyone who has a suggestion about litterature that could calculate the energy in the irst threads energy formula a bit more precisely?

The volume obtained when using the classical electron radius is very close to the volume of the 1s orbital. Any comments about problems or further guidance is greately appreciated? In this derivation the charge density is assumed constant in the energy formulain the first post of this thread. But in 1s the charge density is not constant. Anyone who has a suggestion about litterature that could calculate the energy in the irst threads energy formula a bit more precisely?

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