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I am asked to make the near zone approximation instead of the far zone (radiation zone) approximation, that is to assume kr<<1 instead of kr>>1 for both the magnetic and electric fields.
We are told that the B and E field before making the near zone approximation is given by:
[tex]\vec{B}=k^2 \frac{e^{ikr}}{r} \left( 1+\frac{i}{kr} \right) \hat{r} \times \vec{p_{\omega}}[/tex]
[tex]\vec{E}=\frac{i}{k} (\vec{\nabla} e^{ikr}) \times \left( \frac{k^2}{r} \left(1+\frac{i}{kr} \left) \hat{r} \times \vec{p_{\omega}} \left) + \frac{i}{k} e^{ikr} \vec{\nabla} \times \left( \frac{k^2}{r} \left( i +\frac{i}{kr} \right) \hat{r} \times \vec{p_{\omega}} \right) [/tex]
I am not sure where to start on the approximation where kr<<1, would it just mean ignoring terms that are not 1/(kr)? Since 1/(kr) will be very large, we would get something like:
[tex]\vec{B}=\frac{ik}{r} \frac{e^{ikr}}{r} \hat{r} \times \vec{p_{\omega}}[/tex]
[tex]\vec{E}=\frac{i}{k} (\vec{\nabla} e^{ikr}) \times \left( \frac{ik}{r^2} \hat{r} \times \vec{p_{\omega}} \left) + \frac{i}{k} e^{ikr} \vec{\nabla} \times \left( \frac{ik}{r^2} \hat{r} \times \vec{p_{\omega}} \right) [/tex]
I also need to show that |B|<<|E|, but is what I have above correct?
We are told that the B and E field before making the near zone approximation is given by:
[tex]\vec{B}=k^2 \frac{e^{ikr}}{r} \left( 1+\frac{i}{kr} \right) \hat{r} \times \vec{p_{\omega}}[/tex]
[tex]\vec{E}=\frac{i}{k} (\vec{\nabla} e^{ikr}) \times \left( \frac{k^2}{r} \left(1+\frac{i}{kr} \left) \hat{r} \times \vec{p_{\omega}} \left) + \frac{i}{k} e^{ikr} \vec{\nabla} \times \left( \frac{k^2}{r} \left( i +\frac{i}{kr} \right) \hat{r} \times \vec{p_{\omega}} \right) [/tex]
I am not sure where to start on the approximation where kr<<1, would it just mean ignoring terms that are not 1/(kr)? Since 1/(kr) will be very large, we would get something like:
[tex]\vec{B}=\frac{ik}{r} \frac{e^{ikr}}{r} \hat{r} \times \vec{p_{\omega}}[/tex]
[tex]\vec{E}=\frac{i}{k} (\vec{\nabla} e^{ikr}) \times \left( \frac{ik}{r^2} \hat{r} \times \vec{p_{\omega}} \left) + \frac{i}{k} e^{ikr} \vec{\nabla} \times \left( \frac{ik}{r^2} \hat{r} \times \vec{p_{\omega}} \right) [/tex]
I also need to show that |B|<<|E|, but is what I have above correct?