$13\mathrm{.}5($a$)$ Instead of a constant current distribution assumed for the short dipole of Section $13\mathrm{.}2,$ assume a triangular current distribution $I_s=I_o(1-\frac{2|z|}{l})$ shown in Figure $13\mathrm{.}27.$

Show that

$R_{\text{r}\text{a}\text{d}}=20{}^2[\frac{l}{}{]}^2$

which is one$-$fourth of that in eq$.(13\mathrm{.}13).$ Thus Rrad depends on the current distribution.

$($b$)$ Calculate the length of the dipole that will result in a radiation resistance of $0\mathrm{.}5$ V $.$

Figure $13\mathrm{.}27$ Short dipole antenna with triangular current distribution; for Problem $13\mathrm{.}5.$