General dipole radiation far field equation

In summary, the conversation discusses the search for a reference describing the far-field electric and magnetic field of a dipole. However, the individual is looking for a general formula for an arbitrary scalar function, rather than the typical formula assuming a continuous sinusoid. The conversation also mentions a desire for a transient in time, specifically for a spatial dipole that is pulsed on and off. The general solution for a current source is given, and it is mentioned that the continuity condition is mandatory in order to avoid violating gauge invariance.
  • #1
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I am looking for a reference describing the far-field electric and magnetic field of a dipole. However, I want a general formula for an arbitrary scalar function, and not specifically the usual formula which assumes a continuous sinusoid:

$$ \mathbf{B} = -\frac{\omega^2 \mu_0 p_0 }{4\pi c} \sin\theta \frac{e^{i\omega (r/c-t)}}{r} \mathbf{\hat{\phi} } $$
$$ \mathbf{E} = c \mathbf{B} \times \hat{\mathbf{r}}
= -\frac{\omega^2 \mu_0 p_0 }{4\pi} \sin\theta \frac{e^{i\omega (r/c-t)}}{r} \hat{\theta} $$

I would like something similar, but where I could use say a square pulse, or a Gaussian pulse, or some other non-oscillating waveform.
 
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  • #2
This is not so simple. The reason is that these far-field equations are indeed describing the non-transient quasi-stationary states of the field of one harmonic mode of the electromagnetic field, i.e., it describes the field far from the (compact) source a long time after it was "switched on". For a wave packet, it's not clear to me what you want to achieve since, a wave packet of finite temporal extension, at one point far from the source, this wave packet will run through (more or less deformed by dispersion if it's not a plane wave in vacuo) and the non-transient limit is then simply that there is no field there anymore, because the wave packet as moved further.

Of course you can always use the multipole expansion to build any wavepacket out of the harmonic modes you like via a Fourier integral.
 
  • #3
Yes, I want the transient in time, but I am only interested in the spatial dipole. I want the field of a dipole which is pulsed on and then back off.
 
  • #4
Then you can just take the Fourier transform of the dipole fields, you've written down above, i.e.,
$$(\vec{E}(t,\vec{x}),\vec{B}(t,\vec{x}))=\int \mathrm{d} \omega/(2 \pi) \tilde{A}(\omega) (\vec{E}_{\omega}(t,\vec{x}),\vec{B}_{\omega}(t,\vec{x})).$$
The ##\tilde{A}(\omega)## is given by the Fourier transform of the initial conditions for the fields.
 
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  • #5
Vanhees's equation might be applicable. But if I were in your position I would go back to the starting point where you have to model a dipole which oscillates non-sinusoidally and express this oscillating current in frequency domain, which is the spectrum. Then stay in frequency domain during the derivation since dealing with linear differentiation in frequency domain is easier. Note that those expressions of E and B fields are derived under several assumptions concerning the distance, dipole length, and wavelength, you would simply need to replace the wavelength with the bandwidth of the pulse in any appearing inequality.
 
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  • #6
The general retarded potential solution for a current source ## \mathbf{J}(\mathbf{r},t) ## is,
## \mathbf{A}(\mathbf{r},t) = \frac{\mu_0}{4 \pi} \int \frac{d^3\mathbf{r}^\prime \, \, \mathbf{J}\left(\mathbf{r}^\prime, t-\left|\mathbf{r}-\mathbf{r}^\prime\right|/c\right)}{ \left|\mathbf{r}-\mathbf{r}^\prime\right|} ##

If I use a current density source of ## \mathbf{J}(\mathbf{r},t) = \hat{\mathbf{z}} I d\ell dt \delta(\mathbf{r}) \delta(t) ## then for the z component of the vector potential I get
## A_z(\mathbf{r},t) = \frac{\mu_0 I d\ell dt }{4\pi r}\delta(r-ct) ##

This result makes intuitive sense, so I think it is okay. This can also be written in spherical coordinates,

## A_\theta(\mathbf{r},t) = -\frac{\mu_0 I d\ell dt }{4\pi r}\delta(r-ct) \sin\theta##
## A_r(\mathbf{r},t) = \frac{\mu_0 I d\ell dt }{4\pi r}\delta(r-ct) \cos\theta##

jason
 
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  • #7
I realized I had a careless error in my post (argument of delta function wrong). It should read

##A_z(\mathbf{r},t) = \frac{\mu_0 I d\ell dt }{4\pi r}\delta(t-r/c)##
and likewise for the components in spherical coords.

Also, it is hopefully clear that a more general source,
## \mathbf{J}(\mathbf{r},t) = \hat{\mathbf{z}} I d\ell \delta(\mathbf{r}) f(t)##
will yield,
##A_z(\mathbf{r},t) = \frac{\mu_0 I d\ell }{4\pi r}f(t-r/c)##
As always, the spherical componeents of ##\mathbf{A}## are then,
## A_\theta = -A_z \sin\theta ##
## A_r = A_z \cos\theta ##
And the fields would be derived from ## \mathbf{B} = \nabla \times \mathbf{A}##, etc.

jason
 
  • #8
jasonRF said:
Also, it is hopefully clear that a more general source,
##J(r,t)=z^Idℓδ(r)f(t) \mathbf{J}(\mathbf{r},t) = \hat{\mathbf{z}} I d\ell \delta(\mathbf{r}) f(t)##
This seems like what I am looking for. Do we need to worry about charge density and continuity?

I was pursuing a similar approach, but enforcing continuity and I started getting derivatives of delta functions.
 
  • #9
Charge continuity is included here, and you can calculate the charge density as a function of time.

I have been assuming Lorentz gauge that connects the vector and scalar potential, ##\nabla \cdot \mathbf{A(\mathbf{r},t)} = -\mu \epsilon \frac{\partial \phi(\mathbf{r},t)}{\partial t}##. So once you know ##\mathbf{A}## you can integrate to get ##\phi##, and then calculate the charge density from ##\nabla^2\phi - \mu \epsilon \frac{\partial^2 \phi(\mathbf{r},t)}{\partial t^2} = -\frac{\rho}{\epsilon}##. Or equivelantly, you calculate ##\mathbf{E} = -\nabla \phi - \frac{\partial \mathbf{A}}{\partial t}## and then get the density from ##\rho = \epsilon \nabla \cdot E ##. It can be instructive to look at a simple dipole this way.

EDIT: I was being slow here. Just use the continuity equation to ghet ##\rho## ! ##\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}##

jason
 
  • #10
Yep, the continuity condition is mandatory. Otherwise you violate gauge invariance, and this is a no-go. With ##c \neq 1## it reads
$$\partial_{\mu} j^{\mu}=\frac{1}{c} \partial_t (c \rho) + \vec{\nabla} \cdot \vec{j}=\partial_t \rho+\vec{\nabla} \cdot \vec{j}.$$
So if you give ##\vec{j}## arbitrarily, you have indeed to choose ##\rho## as given in Posting #9.
 

Related to General dipole radiation far field equation

What is the general dipole radiation far field equation?

The general dipole radiation far field equation is a mathematical representation of the electromagnetic radiation emitted by a dipole antenna. It describes the electric and magnetic fields at a distance from the antenna in terms of the dipole's physical characteristics, such as its length and current.

How is the general dipole radiation far field equation derived?

The equation is derived from Maxwell's equations, which govern the behavior of electromagnetic fields. It is also based on the dipole approximation, which assumes that the antenna is much smaller than the wavelength of the emitted radiation.

What factors influence the values in the general dipole radiation far field equation?

The values in the equation are influenced by various factors, including the frequency of the radiation, the distance from the antenna, the orientation of the antenna, and the properties of the surrounding medium.

What are the applications of the general dipole radiation far field equation?

The equation is commonly used in the design and analysis of dipole antennas, which are widely used in telecommunications and radio frequency systems. It also has applications in fields such as radar, satellite communications, and wireless power transmission.

What are the limitations of the general dipole radiation far field equation?

While the equation is a useful tool for understanding and predicting the behavior of dipole antennas, it has some limitations. For example, it assumes that the antenna is in free space and does not take into account the effects of other nearby structures or objects. It also only applies to antennas with a simple, linear geometry and may not accurately predict the behavior of more complex antennas.

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