- #1

- 142

- 4

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).

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- Thread starter qnach
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In summary, Jackson discussed two center-fed linear antenna: one at page 412 (section 9.2) and the other at page 416 (section 9.4). The difference between the two 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.

- #1

- 142

- 4

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).

- #2

- 24,155

- 14,653

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.

- #3

- 142

- 4

vanhees71 said:

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.?

- #4

- 142

- 4

vanhees71 said:

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.?

- #5

- 24,155

- 14,653

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 said: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.?

- #6

- 142

- 4

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)

- #7

- 24,155

- 14,653

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!

- #8

- 142

- 4

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?

A center-fed linear antenna is a type of antenna that consists of a vertical or horizontal wire or rod that is fed at its center. It is typically used for transmitting or receiving electromagnetic waves in the radio frequency range.

A center-fed linear antenna works by converting electrical energy into electromagnetic radiation. The center feed allows for the antenna to be symmetrically balanced, which helps to reduce the amount of interference and improve the efficiency of the antenna.

One advantage of using a center-fed linear antenna is that it has a relatively high gain, which means it can transmit or receive a strong signal. It also has a wide bandwidth, allowing it to operate over a range of frequencies. Additionally, it is easy to construct and has a simple design.

One limitation of a center-fed linear antenna is that it is sensitive to the environment and can be affected by nearby objects such as buildings or trees. It also has a narrow radiation pattern, meaning it is not ideal for omnidirectional use. In addition, it may require a larger amount of space compared to other types of antennas.

A center-fed linear antenna is different from other types of antennas in that it has a symmetrical design, with the feed point at the center. This allows for better balance and performance compared to asymmetrical designs. It also has a narrower radiation pattern compared to omnidirectional antennas, making it more suitable for directional communication.

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