# Distribution of current, charge and EM radiation from a centrally AC fed ball-shaped antenna

As far as I can understand, it is generally accepted that every classical electromagnetic phenomena can be explained by five equations: Maxwell's four equations and the Lorentz force law. However you reply to my question, please start from these equations .

Consider having a perfectly conducting ball of known radius $R$ that at its center point is fed with a known alternating current source $$I = I_o \cos(ωt).$$ Using Maxwell equations we can easily find that: \begin{align} \text{current-distribution} \equiv& \vec{J} = {\vec{J}}_{r}(r) = -\frac{σ_{ο}R^{2}}{r^{2} }θ(-r+R) ω \cos(ω) \hat{r}\\ \text{charge-distribution} \equiv& ρ = σ_{ο} \sin(ωt) δ(r-R)\\ \text{total charge inside the ball} \equiv& Q(t)=σ_{o}4πR^{2}sin(ωt) \end{align} Solving the wave equation for the electric potential $Φ$ in the Lorenz gauge I get: $$\vec{E} = {\vec{E}}_{r}(r,t) = \left(kq\frac{\cos(kr-ωt)}{r} - q\frac{\sin(kr-ωt)}{r{^2}}\right)\hat{r} \quad\text{ where } r>R$$ This wave is a longitudinal wave in vacuum. It can be easily proved that Maxwell equations predict only transverse electromagnetic waves in vacuum. To make this problem more confusing there will be no magetic filed component since there is no transverse current density . What am I doing wrong then?

Practically a centrally fed with electric current ball-shaped antenna can created by digging a straight line to the center of the ball and putting in that line a cable that is insulated everywhere except at its end point that touches the center of the ball. The other end point of the cable is connected to the alternating current source. The alternating current source is earthed.

• I think you just need conservation of charge and spherical symmetry. – Keith McClary Sep 30 '15 at 3:44
• Thanks for the criticism and help about my problem . I will edit the topic so I can make clearly the points I do not understand . – liaguridio Sep 30 '15 at 13:45