Half Wave Antenna Vector Potential

[stunner5000pt]

Vector Potential

Homework Statement

Consider two half wave antennas each ahving current

$$I(z,t)=\hat{z} I_{0}\cos\omega t\sin k(\frac{d}{2}-|z|)$$
where $k=\omega/c$
Each antenna has length d and points in the z direction. Antenna 1 is at $(\Delta/2,0,0)$ and antenna two is at $(-\Delta/2,0,0)$

a) Find the vector potential A
b) Find the electric and magnetic field
c) Find $$dP/d\Omega$$
d) Evalute $$dP/d\Omega$$ in the X Y plane when the antenna is seaparated by a distance lambda/2. Along what direction is the radiation preferentialy propagated?

Homework Equations

In CGS units so...
$$\vec{A}(\vec{r},t)=\frac{1}{c}\int \frac{\vec{J}(\vec{r},t_{r})}{|\vec{r}-\vec{r'}|} d\tau$$

The Attempt at a Solution

So we need the current as a function of z' and the retarded time

$$I(z,t)=\hat{z} I_{0}\cos\omega t\sin k(\frac{d}{2}-|z|)$$
$$I(z,t)=\hat{z} I_{0}\cos\omega (t-\frac{\mathcal{R}}{c})\sin k(\frac{d}{2}-|z|)$$

where $$\mathcal{R}=\sqrt{z'^2+r^2-2z'r\cos\theta}$$
since we want the fields far away (radiation zone), expand
$$\mathcal{R}\approx r\left(1-\frac{z'}{r}\cos\theta$$
so then
$$\cos\omega (t-\frac{\mathcal{R}}{c})\approx\cos\omega\left(t-\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)\right)$$
$$\cos\omega (t-\frac{\mathcal{R}}{c})\approx\cos\omega t \cos\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)+\sin\omega t\sin\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)$$

So then A is calculated like this? make the approximation that $$\mathcal{R}\approx r$$

$$\vec{A} = \hat{z}\frac{I_{0}}{rc}\int \sin k\left(\frac{d}{2}-|z|\right)\left(\cos\omega (t-\frac{\mathcal{R}}{c})\left(\cos\omega t \cos\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)+\sin\omega t\sin\frac{r}{c}\left(1-\frac{z'}{r}\cos\theta\right)\right) dz'$$

Ok since there are two antennas how should the integration be performed...
should i do for each antenna separately? That is integrate one of them from $\Delta$ to $\Delta+\frac{d}{2}$ and one of them from $\Delta$ to $\Delta-\frac{d}{2}$ ?? And then add the two results?

Thanks for your help!

 

[Meir Achuz]

Since it is a radiation problem, I think you are allowed to assume r>>r' and
kr>>1, which greatly simplifies the problem.