Electrostatics
Electricity at rest: charges, the forces between them, field and potential. This is where the capacitor comes from, the component found in every power supply, every filter and every oscillator in the rest of the program.
01Electric charge
Charge is a property of matter, like mass. It comes in two signs and obeys three principles:
- There are two kinds. Positive (proton) and negative (electron). Like charges repel, unlike charges attract.
- Charge is conserved. It is neither created nor destroyed: it is only transferred from one body to another. When something becomes negatively charged, something else is left positive.
- Charge is quantized. Every value of charge is an integer multiple of the elementary charge e = 1.602 × 10−19 C. There are no half charges.
One coulomb is an enormous quantity: it equals 6.25 × 1018 electrons. That is why the charges in electrostatics are measured in microcoulombs or nanocoulombs.
Ways to charge a body
Two different materials are rubbed together and one pulls electrons from the other. This is the mechanism behind the ruler that attracts bits of paper and the spark when you get out of the car.
A charged body touches another and transfers charge to it. Both end up with charge of the same sign.
A charged body is brought near without touching and redistributes the charges of the other. If the other body is connected to ground at that moment and then the charged body is removed, it is left charged with the opposite sign.
02Coulomb's law
The force between two point charges is proportional to the product of the charges and inversely proportional to the square of the distance.
Coulomb's law has the same form as the law of universal gravitation, but it is incomparably stronger: between two electrons, the electric repulsion is about 1042 times greater than the gravitational attraction. If matter were not almost perfectly neutral, electric forces would dominate everything.
Two charges of +2 µC and −3 µC separated by 5 cm. Calculate the force.
An attractive force (opposite signs) of 21.6 N: equivalent to the weight of more than 2 kg, between two charges of a few microcoulombs. And if the distance were doubled, the force would fall to one quarter (5.4 N), not to one half: that is what “inversely proportional to the square” means.
03Electric field
Instead of thinking of forces “at a distance”, it is better to say that a charge modifies the space around it. That modification is the electric field, and another charge placed there feels a force.
04Potential and potential difference
Moving a charge within an electric field costs (or delivers) work. The potential at a point is that work per unit of charge.
What is measured in practice is not the potential of a point but the potential difference between two points, which is what we call voltage. Speaking of the “potential of a point” implies choosing an arbitrary reference: in a circuit, the ground; in an installation, the earth.
These are the places where the potential is the same. Moving a charge along an equipotential costs no work. Equipotentials are always perpendicular to the field lines. A conductor in equilibrium is entirely an equipotential: that is why the metal case of a piece of equipment, if properly connected, has the same potential at all its points.
Two parallel plates 2 mm apart with 12 V between them:
E = 12 V / 0.002 m = 6000 V/m = 6 kV/m.
With the same voltage but a 0.1 mm separation (the typical dielectric thickness of a ceramic capacitor), the field rises to 120 kV/m. That is why a low-voltage capacitor breaks down if more than its rated voltage is applied: it is not the voltage itself, it is the field inside the dielectric.
05Capacitance and the capacitor
A capacitor is two conductors facing each other, separated by an insulator. When voltage is applied, one plate accumulates positive charge and the other the same amount of negative charge. The capacitance measures how much charge it stores per volt applied.
The farad is an enormous unit: common capacitors range from picofarads to millifarads. The formula gives the three ways to increase capacitance:
- More area. That is why electrolytics are two long strips rolled up.
- Less separation. Limited by the voltage it has to withstand.
- A better dielectric (higher εr).
Dielectrics
| Material | εr | Dielectric strength (kV/mm) | Use |
|---|---|---|---|
| Vacuum / air | 1.0 | 3 | Variable tuning capacitors |
| Paper | 3.5 | 16 | Motor-start capacitors |
| Polyester (Mylar) | 3.2 | 300 | General use in electronics |
| Polypropylene | 2.2 | 400 | Audio, high frequency, low losses |
| Mica | 5.4 | 100 | High stability, radio frequency |
| Ceramic NP0/C0G | 30 | — | Oscillators: very stable with temperature |
| Ceramic X7R / Z5U | 1000 to 10000 | — | Decoupling: high capacitance, poor stability |
| Aluminum oxide (electrolytic) | 8 | 700 | Filtering. The oxide layer is nanometers thick: hence the enormous capacitance |
Every insulator breaks down if the field exceeds a certain value. That limit is the dielectric strength, and from it comes the working voltage printed on the capacitor. Exceeding it destroys it, sometimes explosively. Rule of thumb: choose a capacitor rated for at least twice the voltage it will have to withstand, and in circuits connected to the mains, much more.
The capacitors on the shop bench
| Type | Range | Polarized? | Characteristics |
|---|---|---|---|
| Ceramic | 1 pF to 1 µF | No | Cheap and small. The 100 nF one is the universal decoupling capacitor. |
| Polyester | 1 nF to 10 µF | No | Stable, good tolerance. Audio coupling. |
| Aluminum electrolytic | 1 µF to 47 mF | Yes | High capacitance, poor tolerance (−20/+80 %), dries out over the years. |
| Tantalum | 0.1 to 100 µF | Yes | Smaller and more stable than aluminum, but it catches fire if connected backwards. |
| Variable / trimmer | 5 to 500 pF | No | Radio tuning. |
| Supercapacitor | 0.1 to 3000 F | Yes | Low voltage (2.7 V). Memory backup, starting. |
Ceramic and polyester capacitors are marked with a three-digit code in picofarads: the first two are the significant digits and the third is the number of zeros. So, 104 = 100,000 pF = 100 nF = 0.1 µF.
🔤 Capacitor code decoder and SI converter Translates the 3-digit code into pF, nF and µF (including the cases 8 = ×0.01 and 9 = ×0.1 and the R notation) and converts between all the prefixes. ›06Capacitors in series and in parallel
They behave the opposite way from resistors, and that reversal is the cause of half the errors in the exercises.
It is like increasing the area of the plates. All of them have the same voltage and the working voltage of the assembly is that of the lowest.
It is like increasing the separation. All of them have the same charge and the voltages add: that is how you get to withstand more voltage.
CT = 235 µF and the allowable voltage becomes 400 V. This is what is done in the power supplies of large audio equipment. Careful: in practice you have to add equalizing resistors (100 kΩ) in parallel with each capacitor, because the leakage currents differ and, without them, the voltage is shared poorly and one of the two ends up withstanding almost all of it.
07Energy and the RC circuit
A capacitor does not charge instantaneously: if it is connected to a source through a resistor, the charge follows an exponential curve governed by the time constant.
This behavior is the basis of a lot of circuits seen later: the power supply filter, timers with the 555, debounce circuits and low-pass filters.
⏱️ LM555 simulator Shows live the charging and discharging of the capacitor through the resistors, with the animated charge level and the timing calculation on screen. ›08Electrostatics in the shop: ESD
This is not an anecdotal subject: electrostatic discharge (ESD) is one of the main causes of component destruction in the electronics industry.
| Situation | Typical voltage generated |
|---|---|
| Walking on a carpet (dry air) | up to 35,000 V |
| Walking on a vinyl floor | up to 12,000 V |
| Getting up from a plastic chair | up to 18,000 V |
| Opening an ordinary plastic bag | up to 20,000 V |
| Minimum a person can feel | ≈ 3,000 V |
| Enough to damage a CMOS device | ≈ 100 V |
You can destroy an integrated circuit with a discharge that the person does not even notice. And often the damage is latent: the component keeps working, but fails weeks later. That is why you work with an antistatic wrist strap grounded through 1 MΩ (the resistor protects the person), a dissipative mat, and by keeping CMOS ICs in conductive foam or a metallized bag. It is especially critical with the 40xx series and with microcontrollers.
09In the lab
With a leaf electroscope: (a) rub a plastic rod with wool and bring it near without touching — the leaves separate by induction and come back together when it is moved away; (b) touch the electroscope with the rod — it charges by contact and the leaves stay apart; (c) touch the electroscope with a finger to discharge it to ground. Note the sign of the charge in each case.
With a multimeter that has a capacitance function, measure three different polyester capacitors and compare them with their printed code (checking whether they are within tolerance). Then connect them in parallel (measure: it should give the sum) and in series (measure: it should give less than the smallest). It is the exercise that fixes the reversal with respect to resistors.
R = 100 kΩ, C = 100 µF (τ = 10 s, slow on purpose so you can see it with the multimeter). Power it with 9 V and note the capacitor voltage every 2 seconds up to 50 s. Plot it and verify that at 10 s the voltage is 63 % of 9 V, that is 5.67 V. Repeat for the discharge by shorting the source and check that the curve is the mirror image.
Charge a 4700 µF electrolytic to 12 V and calculate the energy: W = ½ × 0.0047 × 144 = 0.34 J. Discharge it through a 12 V lamp and watch the flash. Never discharge an electrolytic by shorting it with a screwdriver: the peak current is tens of amperes and it can throw out molten metal.
A large charged electrolytic holds its charge for hours after the equipment is disconnected. In mains-powered supplies that charge is at 310 V and is dangerous. Before touching any circuit board: disconnect, wait, discharge with a 1 kΩ / 5 W resistor and only then verify with the voltmeter that it reads zero.
10Common errors
| Symptom | Usual cause |
|---|---|
| The electrolytic swelled or exploded | Reversed polarity, or working voltage exceeded. Remember that the peak of an AC voltage is 1.41 times its RMS value. |
| The series capacitor calculation gives more than the individual ones | The resistor formula was applied. For capacitors it is the other way around. |
| The measured capacitance is much lower than the printed one | Old, dried-out electrolytic. It is the most common failure in equipment more than ten years old. |
| A “new” CMOS IC does not work | ESD damage during handling. It may have been invisible. |
| The RC circuit takes much longer than calculated | Units: R in kΩ with C in µF gives milliseconds, not seconds. |
| A tantalum capacitor caught fire | It was mounted backwards. Tantalum capacitors mark the positive terminal with the band, the opposite of aluminum electrolytics, which mark the negative. |
11Self-assessment
If the distance between two charges is doubled, what happens to the force?
It is reduced to one quarter. Coulomb's law is inversely proportional to the square of the distance.
What is the difference between electric field and electric potential?
The field is force per unit of charge (a vector quantity, in N/C or V/m); the potential is energy per unit of charge (a scalar quantity, in volts). In a uniform field they are related by E = V/d.
A 100 µF capacitor charged to 12 V: how much charge does it store and how much energy?
Q = C·V = 100 µF × 12 V = 1.2 mC.
W = ½·C·V² = 0.5 × 0.0001 × 144 = 7.2 mJ.
How do you increase the capacitance of a parallel-plate capacitor?
By increasing the facing area, reducing the separation or using a dielectric with a higher relative permittivity. The trade-off of reducing the separation is that the working voltage goes down.
Three 30 µF capacitors: what is the total capacitance in series and in parallel?
In parallel: 30 + 30 + 30 = 90 µF.
In series: 1/C = 3/30 → C = 10 µF. With n equal capacitors in series, the total is the
value divided by n.
What does the code “223” on a ceramic capacitor mean?
22 followed by 3 zeros, in picofarads: 22,000 pF = 22 nF = 0.022 µF.
R = 10 kΩ and C = 47 µF. How long does it take to charge practically fully?
τ = 10 kΩ × 47 µF = 470 ms. It is considered charged at 5τ, that is 2.35 seconds.
Why does the antistatic wrist strap include a 1 MΩ resistor?
To protect the person. The resistor lets the static charge drain away (which carries very little energy) but limits the current to a safe value if by accident a point at mains voltage is touched. A wrist strap connected directly to ground would be dangerous.
Can a discharge that you cannot feel damage a component?
Yes. A person starts to perceive a discharge at around 3000 V, while a CMOS IC can be damaged at 100 V. Also, the damage is often latent: the component keeps working and fails later on.