I am reading Classical Electricity And Magnetism by Wolfgang K. H. Panofsky and Melba Phillips, and in particular the section on antennae in chapter 14.

I understand the current in the antenna should vanish at the end points and have a structure of standing wave. The problem is how to derive equation 14.54:

$$ \mathbf{j}_0 (x',y',z') = J_0 ~ \delta(y')\delta(z') \frac{\sin[k(L/2 - |x'|^2)}{\sin(kL/2)} $$

Does it have to do something solution of sound waves in organ pipe or what? Please explain. $L/2$ is the length of antenna from central feed.

  • $\begingroup$ The formula looks dimensionally incorrect. $\endgroup$
    – Farcher
    Jul 7, 2022 at 8:28

1 Answer 1


According to the Balanis's book (C.A. Balanis, "Antenna theory- Analysis and design", John Wiley and Sons Inc), the sinusoidal form is experimentally verified fact. As long as I know, it seems there is no mathematical proof.

The following statements are taken from Balanis' book:

4.5.1 Current Distribution

For a very thin dipole (ideally zero diameter), the current distribution can be written, to a good approximation, as \begin{equation*} \mathbf{I}_l(x'=0,y'=0,z')= \begin{cases} \hat{\mathbf{a}}_zI_0\text{sin}\left[k\left(\frac{l}{2}-z'\right)\right],\;\;\;(0\leq z'\leq l/2)\\ \hat{\mathbf{a}}_zI_0\text{sin}\left[k\left(\frac{l}{2}+z'\right)\right],\;\;\;(-l/2 \leq z'\leq 0) \end{cases} \text{ (4-56)} \end{equation*} This distribution assumes that the antenna is center-fed and current vanishes at the end points ($z'=\pm l/2)$. Experimentally, it has been verified that the current in a center-fed wire antenna has sinusoidal form with nulls at the end points. For $l=\lambda/2$ and $\lambda/2<l<\lambda$ the current distribution (4-56) is shown plotted in Figures 1.16(b) and 1.12(c), respectively. The geometry of the antenna is that shown in Figure 4.5


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