# Electric field boundary equation implication at air/earth interface

Hi,

In the case of dielectric in electric field E0, we have E1 = E0 outside crystal and E2 = E0 - P inside, so that D1 = E1 = D2 = E2 + P. That's obvious.

Now, if dielectric is ferroelectric and no external field, we have only bounded surface charges. It seems for me that E1 = 0 outside crystal and E2 = - P inside. So we have D1 = 0 outside and D2 = E2 + P = 0.

But I've read (link ), that polarization close to the free surface and perpendicular to it must vanish in order to fulfill boundary condition from Maxwell equations that D1 - D2 = Qfree. Since there are no free charges, we have D1 = D2. But if we have D1 = 0 and D2 = 0, I don't see any restriction for P...?

Then, if we have a metal on a ferroelectric surface (the same reference), rather than vanishing of polarization, we have free charges induced on the dielectric-metal boundary to fulfill this boundary condition. Thus... I don't know!

It looks very strange for me. How to understand it?

I don't think this counts as homework, if it does then apologies!

I'm doing some simulation work on the structure of a CCD and noticed something peculiar. If i have an electrode, seperated from a substrate material by a dielectric, and i bias it by 12V then i get a potential underneath it- as i expect. However if i increase the size of the electrode (read as area) then the value of the maximum potential (VMAX) increases. Its position also moves further away from the electrode itself.

I've been trying to understand why, as in thoery the charge density remains constant, so although the extent of the field would increase, i can't understand why the value of the maximum changes.

Since the field is the derivative of the potential, a change in VMAX would imply that that field itself is changing in magnitude (read as flux density) as well.

Any ideas? Perhaps my logic is flawed, if so, then please point it out as i'd like to understand it.

If we consider an arbitrary time varying current $I\left(t\right)$ flowing through an arbitrary antenna, then is the following a plausible expression for the electric field $\vec{\mathrm{E}}\left(\vec{\mathrm{r}},t\right)$ in the far field region:
$\displaystyle\vec{\mathrm{E}}\left(\vec{\mathrm{r}},t\right) = \frac{1}{4\pi \epsilon_0 c^2 r} \; \sum_{i=1}^{n} \left[ \vec{\mathrm{a}}_i\left(\hat{\mathrm{r}}\right) \; \frac {d^i}{dt^i}I\left(t - \frac {r} {c}\right) \right]$​
where the integer value $n$ and all vector functions $\vec{\mathrm{a}}_i\left(\hat{\mathrm{r}}\right)$ depend only on the antenna geometry (i.e. not on any characteristic of the input excitation), and $\vec{\mathrm{a}}_i\left(\hat{\mathrm{r}}\right) \cdot \hat{\mathrm{r}} = 0$ for all $i$?

And if this can indeed be done, what kind of expression can be deduced when the same antenna is receiving radiation -- converting the electric field into a current or voltage?

I am trying to understand if it is possible to characterize an arbitrary arrangement of conductors as an antenna purely in the time domain -- as opposed to parameters like radiation resistance, gain, antenna aperture and radiation pattern that depend on frequency/wavelength.

Thanks...

Spinnor
Gold Member
Suppose we have two co-linear strings under tension, attached far away, and with propagation speed v. Let the strings have equal and opposite linear charge densities. In each hand you hold an end of a string. With the two strings held close, your hands quickly separate and then come back together again. Such action will produce a wave of charge polarization that moves with speed v away from you. Sketch roughly the electric and magnetic fields of the polarization pulse near the pulse. As v << c, c the speed of light explain why the electric field energy density is much greater than the magnetic field energy density.

Now assume the strings have a propagation speed of c the velocity of light, and assume that the physical pulse height is very small with the linear charge densities increased to compensate for the smaller amplitude. Sketch roughly the electric and magnetic fields of a linear polarization pulse moving away from you at light speed.

If we had the right density of a swarm of linear polarization pulses (both up and down) moving mostly in phase and away from a source could the resultant fields of many polarization pulses approximate that of a beamed antenna? Imagine many pairs of such strings above with pulses.

Is the following close for the first case?

Thanks for any help!

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