Making Near Zone Approximation for B and E Fields

[LocationX]

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:

$$\vec{B}=k^2 \frac{e^{ikr}}{r} \left( 1+\frac{i}{kr} \right) \hat{r} \times \vec{p_{\omega}}$$

$$\vec{E}=\frac{i}{k} (\vec{\nabla} e^{ikr}) \times \left( \frac{k^2}{r} \left(1+\frac{i}{kr} \left) \hat{r} \times \vec{p_{\omega}} \left) + \frac{i}{k} e^{ikr} \vec{\nabla} \times \left( \frac{k^2}{r} \left( i +\frac{i}{kr} \right) \hat{r} \times \vec{p_{\omega}} \right)$$

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:


$$\vec{B}=\frac{ik}{r} \frac{e^{ikr}}{r} \hat{r} \times \vec{p_{\omega}}$$

$$\vec{E}=\frac{i}{k} (\vec{\nabla} e^{ikr}) \times \left( \frac{ik}{r^2} \hat{r} \times \vec{p_{\omega}} \left) + \frac{i}{k} e^{ikr} \vec{\nabla} \times \left( \frac{ik}{r^2} \hat{r} \times \vec{p_{\omega}} \right)$$

I also need to show that |B|<<|E|, but is what I have above correct?

 

[turin]

What do you think e^ikr approximates to? BTW, don't forget that there is an e^iwt kind of factor on top of all of this, which may not be important for this calculation, but it is important to save confusion.

 

[LocationX]

What do you think e^ikr approximates to? BTW, don't forget that there is an e^iwt kind of factor on top of all of this, which may not be important for this calculation, but it is important to save confusion.

e^ikr when kr<<1 in this approximation is around 1?

What is the e^iwt factor?

 

[turin]

What is the e^iwt factor?

Firstly, I should say that the "w" in that expression is usually written as the lower-case Greek letter, Omega; I am just lazy, and I don't like the way tex is rendered on this forum, anyway. This factor may not be important for this problem, depending on the form you need to put the answer in. If you are supposed to find the actual field values, instead of the phasors, then it is important. In the phasor formalism of E&M, there is always an implied factor of e^iwt, so this effects the real part by carrying an imaginary term that can also multiply the imaginary part of the rest of the expression to give an additional real piece.

For example, if the "k" part of the expression is purely real, then you would have an extra factor of cos(wt), and if it is purely imaginary, then you would have a factor of sin(wt). In general (both real and imaginary parts), this extra phase factor determines a sine or cosine with some phase.

Again, this is unimportant for this problem if you are only supposed to give the phasor "k" part of the expressions. If you want to understand E&M, though, then it is important to realize that the expressions you have there for E and B are not expressions for the physical fields (because, for one thing, they are not real-valued).

I should also warn you, when you approximate the e^ikr, you should always keep (at least) the lowest order term that gives a nonvanishing result, and then, after you have approximated it everywhere in the expressions, sort through and keep the lowest order term(s).