Electrostatics
The damage threshold of a MOSFET starts at about 100 V, while the threshold of human perception is near 3000. A component can be destroyed by a discharge that the person did not feel.
01Coulomb's law and the field
Two charges at rest attract or repel each other with a force that falls with the square of the distance. The form is identical to that of gravitation, with one decisive difference: charge has two signs, and that is why the electric force can cancel.
So that we do not always have to talk about pairs of charges, the electric field is defined: what a distribution of charges does to space, regardless of whether there is anything there to feel it.
Enormous. Two charges of one coulomb one meter apart would repel each other with nine billion newtons, the weight of a million tons. That is why electrostatics works in nanocoulombs and microcoulombs: the charge that a person builds up walking across a carpet is on the order of a microcoulomb, and that is enough to destroy an integrated circuit.
02Flux and Gauss's law
The flux of the field through a closed surface measures how many field lines cross it net, and it turns out to depend only on the enclosed charge:
| Distribution | Field | How it falls off |
|---|---|---|
| Point charge | \( E = kq/r^2 \) | With the square |
| Infinite charged wire | \( E = \lambda/2\pi\varepsilon_0 r \) | With the distance |
| Infinite charged plane | \( E = \sigma/2\varepsilon_0 \) | It does not fall off: it is uniform |
| Inside a conductor in equilibrium | \( E = 0 \) | Always |
The last row is the Faraday cage, and it is the principle behind all electronic shielding. If there were a field inside the metal, the free charges would move; they move until the internal field vanishes, and that happens in nanoseconds. That is why a metal enclosure protects what is inside it, and why the braid of a coaxial cable works.
03Electric potential
Since the electric force is conservative, a potential energy can be associated with it, and dividing it by the test charge gives the potential, which is a scalar and therefore much more convenient than the field:
Tap any point on the map and the field and potential there appear. The color is the potential, the white curves are equipotentials and the arrows are the field. Notice that they always cross at right angles.
04Electrostatic discharge
This whole unit becomes very concrete at the workbench. Walking across a synthetic carpet charges the body to several kilovolts, and when you touch a component that charge discharges through it.
| Situation | Typical voltage | Can you feel it? |
|---|---|---|
| Walking on carpet, dry air | Up to 15 kV | Yes, with a spark |
| Getting up from a chair | 6 kV | Sometimes |
| Handling a plastic bag | 7 kV | Sometimes |
| MOSFET damage threshold | From 100 V | No |
The last row is the one that matters: the damage threshold is one or two orders of magnitude below the perception threshold, which is around 3 kV. A component can be damaged without the person having noticed absolutely anything, and often it does not fail at the moment but weeks later. That is why the antistatic wrist strap, conductive bags and dissipative floors are not a ritual.
05In the lab
In a tray of slightly salted water, place two electrodes connected to a power supply and measure the potential at a grid of points with a voltmeter. Draw the equipotentials and trace the field lines perpendicular to them. Compare with the lab on this page.
Place a radio receiver that is switched on inside a metal can, first open and then closed. Repeat with meshes of different spacing and relate the mesh spacing to the wavelength of the blocked signal.
With an electrometer or a high-impedance multimeter and a reference capacitor, estimate the voltage the body reaches after walking on different floors. Calculate the stored energy and compare it with the damage thresholds in the table.
06Common mistakes
- Confusing field with force. The field exists even if there is no test charge.
- Adding potentials as vectors. Potential is a scalar; it is the field that adds vectorially.
- Believing Gauss's law only holds with symmetry. It always holds; what needs symmetry is being able to solve for \( E \) easily.
- Thinking there can never be a field inside a hollow conductor. If there is charge inside the cavity, there is.
- Assuming that if you did not feel the spark there was no discharge. The damage threshold is far below the perception threshold.
- Drawing field lines that cross. They cannot: the field has only one direction at each point.
07Self-assessment
What does the flux through a closed surface depend on?
Only on the enclosed charge, divided by \( \varepsilon_0 \). Neither on the shape nor on where the charge is inside.
Why is the field inside a conductor in equilibrium zero?
Because if there were a field, the free charges would move. They move until they cancel it.
How does the field of an infinite charged plane fall off?
It does not: it is uniform, equal to \( \sigma/2\varepsilon_0 \) at any distance.
What angle do field lines make with equipotentials?
Ninety degrees, always. The field is the gradient of the potential, with its sign changed.
Why can a MOSFET be damaged without you feeling anything?
Because its damage threshold starts at about 100 V and human perception is near 3000.
How much energy does a 4 kV discharge from the body carry?
About 0.8 mJ, with the typical 100 pF of a person. Little in total, delivered in nanoseconds.
08Further reading
- Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers with Modern Physics. 10th ed., Cengage, 2019 (Spanish edition, “Física para ciencias e ingeniería”). The chapters on electric field, Gauss's law and potential, with plenty of worked problems.
- Edward M. Purcell and David J. Morin. Electricity and Magnetism. 3rd ed., Cambridge University Press, 2016 (Spanish edition, “Electricidad y magnetismo”). The most elegant treatment of the subject there is, with a geometric intuition not found in other books.
- Henry W. Ott. Electromagnetic Compatibility Engineering. Wiley, 2009. Shielding, Faraday cages and protection against electrostatic discharge, applied to real equipment.