Why does Jackson consider small antennas on page 412?

[qnach]

Jackson discussed two center-fed linear antenna:
once at page 412 (section 9.2)
the other at page 416 (section 9.4)
I just cannot figure out what are the difference?
I though they are different method of calculation
but (9.56) is different from (9.28).

 

[vanhees71]

The difference is that in Sect. 9.2 he describes a pure dipol field corresponding to an approximate current density (9.25), while in Sect. 9.4 he describes an exact solution for an antenna of finite length with current density (9.53). One should of course, also think about the necessary condition on charge and current to fulfill the continuity equation to make sure that the ansatz (9.53) is consistent.

Indeed, here we have
$$\rho(t,\vec{x})=\rho'(\vec{x}) \exp(-\mathrm{i} \omega t), \quad \vec{j}(t,\vec{x})=\vec{j}'(\vec{x}) \exp(-\mathrm{i} \omega t).$$
From the continuity equation we have
$$\partial_t \rho+\vec{\nabla} \cdot \vec{j} = (-\mathrm{i} \omega t \rho'(\vec{x})+\vec{\nabla} \cdot \vec{j}'(\vec{x}))\exp(-\mathrm{i} \omega t) \stackrel{!}{=}0.$$
Thus we only have to make
$$\rho'(\vec{x})=-\frac{\mathrm{i}}{\omega} \vec{\nabla} \cdot \vec{j}'(\vec{x}).$$
to fulfill the continuity equation, which is an integrability condition for the Maxwell equations.

See also Sect. 9.12 for the comparison with multipole expansions.

 

[qnach]

The difference is that in Sect. 9.2 he describes a pure dipol field corresponding to an approximate current density (9.25), while in Sect. 9.4 he describes an exact solution for an antenna of finite length with current density (9.53). One should of course, also think about the necessary condition on charge and current to fulfill the continuity equation to make sure that the ansatz (9.53) is consistent.

Indeed, here we have
$$\rho(t,\vec{x})=\rho'(\vec{x}) \exp(-\mathrm{i} \omega t), \quad \vec{j}(t,\vec{x})=\vec{j}'(\vec{x}) \exp(-\mathrm{i} \omega t).$$
From the continuity equation we have
$$\partial_t \rho+\vec{\nabla} \cdot \vec{j} = (-\mathrm{i} \omega t \rho'(\vec{x})+\vec{\nabla} \cdot \vec{j}'(\vec{x}))\exp(-\mathrm{i} \omega t) \stackrel{!}{=}0.$$
Thus we only have to make
$$\rho'(\vec{x})=-\frac{\mathrm{i}}{\omega} \vec{\nabla} \cdot \vec{j}'(\vec{x}).$$
to fulfill the continuity equation, which is an integrability condition for the Maxwell equations.

See also Sect. 9.12 for the comparison with multipole expansions.

1. Why does his (9.53) has no time-dependent part?
2. where does your
$$\rho(t,\vec{x})=\rho'(\vec{x}) \exp(-\mathrm{i} \omega t), \quad \vec{j}(t,\vec{x})=\vec{j}'(\vec{x}) \exp(-\mathrm{i} \omega t).$$
came from? It seems to be a general proof? Everything fufill $\exp(-\mathrm{i} \omega t)$ is o.k.?

 

[qnach]

The difference is that in Sect. 9.2 he describes a pure dipol field corresponding to an approximate current density (9.25), while in Sect. 9.4 he describes an exact solution for an antenna of finite length with current density (9.53). One should of course, also think about the necessary condition on charge and current to fulfill the continuity equation to make sure that the ansatz (9.53) is consistent.

Indeed, here we have
$$\rho(t,\vec{x})=\rho'(\vec{x}) \exp(-\mathrm{i} \omega t), \quad \vec{j}(t,\vec{x})=\vec{j}'(\vec{x}) \exp(-\mathrm{i} \omega t).$$
From the continuity equation we have
$$\partial_t \rho+\vec{\nabla} \cdot \vec{j} = (-\mathrm{i} \omega t \rho'(\vec{x})+\vec{\nabla} \cdot \vec{j}'(\vec{x}))\exp(-\mathrm{i} \omega t) \stackrel{!}{=}0.$$
Thus we only have to make
$$\rho'(\vec{x})=-\frac{\mathrm{i}}{\omega} \vec{\nabla} \cdot \vec{j}'(\vec{x}).$$
to fulfill the continuity equation, which is an integrability condition for the Maxwell equations.

See also Sect. 9.12 for the comparison with multipole expansions.

1. Why does his (9.53) has no time-dependent part?
2. where does your
$$\rho(t,\vec{x})=\rho'(\vec{x}) \exp(-\mathrm{i} \omega t), \quad \vec{j}(t,\vec{x})=\vec{j}'(\vec{x}) \exp(-\mathrm{i} \omega t).$$
came from? It seems to be a general proof? Everything fufill $$\exp(-\mathrm{i} \omega t)$$ is o.k.?

 

[vanhees71]

1. Why does his (9.53) has no time-dependent part?
2. where does your
$$\rho(t,\vec{x})=\rho'(\vec{x}) \exp(-\mathrm{i} \omega t), \quad \vec{j}(t,\vec{x})=\vec{j}'(\vec{x}) \exp(-\mathrm{i} \omega t).$$
came from? It seems to be a general proof? Everything fufill $\exp(-\mathrm{i} \omega t)$ is o.k.?

That's just the usual ansatz for harmonic time dependence. That's pretty general since you can get any wave from it via Fourier transformation. Plugging in the ansatz into Maxwell's equations you get a set of equations for $\vec{E}'$ etc. That's what Jackson just wrote without the prime.

 

[qnach]

How about the first part of my question:

1. Why does his (9.53) has no time-dependent part?

Should everything feed into an antenna have time dependency?
Sorry, the question may sounds trivial for you. But I am really confused.
Jackson seems to use different current for the same antenna? (and one more in section 9-12)

 

[vanhees71]

I answered the question. Jackson writes down everything for the time-independent fields, I labeled with a prime. It's just hist notation.

Of course, if the fields are not time-dependent an antenna is pretty useless, because then you just have static fields, which don't do anything to the antenna, and there are no waves propagating. You cannot send a signal with something not changing in time!

 

[qnach]

I realized on
page 412 Jackson discussed small antenna,
while the other two discussion (page 416 and section 9-12) are all on large antenna.
Jackson states these in his book.

But I cannot see why page 412 need the "small" assumption?