Dielectric Breakdown and the Corona Effect

Let us consider a simple model for an atom. Suppose an atom is a big ball of radius $R$ of uniform negative charge with total charge $-Q$, with a small, heavy, point like positive charge $+Q$ inside. Using Gauss's Law to find the electric field on all space due to the negative charge, we find:

$$\oint_S E_r dA = E_r 4\pi r^2 = -kQ$$

$$E_r = -\frac{kQ}{r^2} \quad\quad (r>R)$$ (23)

outside the ball, and

$$\oint_S E_r dA = E_r 4\pi r^2 = -kQ \frac{r^3}{R^3}$$

$$E_r = -\frac{kQr}{R^3} \quad\quad (r<R)$$ (24)

inside the ball. The force on the ``nucleus'' is thus attractive but getting weaker outside the ball and increases linearly with its displacement from the nucleus on the inside.

Hmmm, given a linear attractive restoring force, we can guess that the nucleus will oscillate harmonically in the negative ball if released from rest on the inside. Cool. Later we'll think about what happens if it is driven harmonically at its resonant frequency. We can also see that if we put the whole ``atom'' in an external electric field, it will polarize, that is, become a dipole. The nucleus will be displaced in the direction of the external field until the net force pulling it back to the origin vanishes.

In a field stronger than $kQ^2/R^2$, though, the nucleus is pushed outside of the ball where the force suddenly drops off. It is ripped away from its negative ball (or vice versa, actually, since the negative electrons are really the lighter things). The point is, that any material, be it insulator or conductor, can be put in a field strong enough to rip its constituent atoms apart. They ionize, and generally when that happens an insulator becomes a very hot conductor. This is the phenomenon know as ``dielectric breakdown''.

Even the best insulators - glass, quartz, mica - will arc across if placed in a big enough field. Insulators, in fact, aren't. They are only insulating in the weak field regime, and tend to cross over from that weak field regime to a conducting state catastrophically. A bolt of lightning, a static spark, St. Elmo's fire, all are the result of an insulating material being torn apart by a strong field so that it conducts. This is one of many reasons high voltage power supplies are so dangerous - 16,000 volts of potential difference will arc right across a gap of a centimeter or so of air. You don't actually have to touch the conductor to die, especially if there are any sharp points nearby.

This is also how lightning rods work. Lightning rods don't attract lightning (which is catastrophically destructive - you do not want to be hit by lightning or to even have lightning hit a lightning rod connected to your house, unless you aren't fond of your house and want to collect the insurance). Lightning rods prevent lightning. Let's see how.

Suppose a big old positively charged up cloud passes by overhead with no lighting rods present. It attracts electrons from the nearby conducting ground, and they pile up directly beneath it. They strain to reach the cloud, running up things like our salty bodies or the trunks of nearby tries to get a bit closer to the positive charge there. However, they cannot easily jump over to the cloud with all that insulating air in the way and the field grows faster than the cloud is neutralized. Eventually the field strength gets to be great enough that the air right above a high point starts to ionize, and BAM, current flows which heats the air which knocks off electrons which conduct current wich flows, further heating the air, reaching ever higher to the cloud. In an instant, a lot of charge flows from the ground to the cloud, which becomes neutralized. The tree, human, air, and anything else granting passage to the huge blast of hot current sizzles gently in its passing, dead, exploded, burnt. Air rushes back into the resulting vacuum, creating the roll of thunder.

NOW suppose there is a lightning rod handy. When the cloud rolls by, electrons are pulled up to the sharp tip of the lightning rod. There even a small surplus charge is enough to cause air around the tip to ionize, and the air around the tip begins to conduct (and will often glow as electrons are knocked about and recombine with atoms). This happens slowly though, slowly enough that the charged air can spray upwards and create a conducting path to the cloud without breaking down all at once or getting so hot. On a good day, the cloud is neutralized without ever discharging a bolt of lightning. On a bad day, lightning hits your lightning rod and it explodes in a shower of molten metal and burns down your house.

Clearly lots of lightning rods are better than just one. So are lots of trees.

Dielectric breakdown and the phenomenon of polarization will be very important to us in the next chapter.

To conclude, may I suggest that you review the methods covered for evaluating the potential from the field (using Gauss's Law, to find the field) as this will be of the greatest use to us in our discussion of capacitance. However, you should all be able to find the potential inside a uniform ball of sphere, for example, as well.

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