- #1
avikarto
- 56
- 9
I am trying to derive that ##\vec{J}=\vec{L}+\vec{S}## (in free space) starting from the general expression ##\vec{J}=\int dV\epsilon_0\: \vec{r}\times\left(\vec{E}\times \vec{B}\right)##. I get to the point where I have that
##\vec{J}=\epsilon_0\int dV E_i\left(\vec{r}\times\nabla\right)A_i+\epsilon_0\int dV \vec{E}\times\vec{A}-\epsilon_0\int dV\left(\vec{E}\cdot\nabla\right)\left(\vec{r}\times\vec{A}\right)##
Equating the first two terms with ##\vec{L}## and ##\vec{S}## respectively, I can't figure out how to make the final term zero. The text I am following indicates that I should use the divergence theorem to make a surface integral vanish, but in order to do so I need to have a divergence, which I don't seem to have. Going through the tensor notation, all I can seem to rearrange this to is
##\left(\vec{E}\cdot\nabla\right)\left(\vec{r}\times\vec{A}\right)=\vec{r}\times\left(\vec{E}\cdot\nabla\right)\vec{A}+\vec{E}\times\vec{A}##
Am I thinking about this the right way? Any direction would be appreciated. Thanks.
##\vec{J}=\epsilon_0\int dV E_i\left(\vec{r}\times\nabla\right)A_i+\epsilon_0\int dV \vec{E}\times\vec{A}-\epsilon_0\int dV\left(\vec{E}\cdot\nabla\right)\left(\vec{r}\times\vec{A}\right)##
Equating the first two terms with ##\vec{L}## and ##\vec{S}## respectively, I can't figure out how to make the final term zero. The text I am following indicates that I should use the divergence theorem to make a surface integral vanish, but in order to do so I need to have a divergence, which I don't seem to have. Going through the tensor notation, all I can seem to rearrange this to is
##\left(\vec{E}\cdot\nabla\right)\left(\vec{r}\times\vec{A}\right)=\vec{r}\times\left(\vec{E}\cdot\nabla\right)\vec{A}+\vec{E}\times\vec{A}##
Am I thinking about this the right way? Any direction would be appreciated. Thanks.
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