1.1 Electric charge

Fundamental properties of charge

Electric charge is the property of matter that causes it to experience a force in an electromagnetic field. It's the starting point for everything in electromagnetism, from how atoms hold together to how circuits work.

Positive vs. negative charge

There are exactly two types of electric charge: positive and negative. The core rule is simple: like charges repel, opposite charges attract.

• Protons carry positive charge
• Electrons carry negative charge
• Neutrons have no net charge

Both protons and electrons carry the same magnitude of charge: $e = 1.602 \times 10^{-19}$ coulombs. The signs are opposite, but the size is identical. This symmetry matters because it's why atoms with equal numbers of protons and electrons are electrically neutral.

Charge quantization

Electric charge doesn't come in any arbitrary amount. It only comes in whole-number multiples of the elementary charge $e$. This is what quantization means: charge is "chunked," not continuous.

• Any observable charge $q$ satisfies $q = ne$, where $n$ is an integer (positive, negative, or zero)
• You'll never measure a free charge of, say, $0.5e$ or $2.3e$

Quarks do carry fractional charges ($+\frac{2}{3}e$ or $-\frac{1}{3}e$), but quarks are permanently confined inside protons and neutrons. They're never observed in isolation, so every measurable free charge is still an integer multiple of $e$.

Charge conservation

The total electric charge in an isolated system never changes. Charge can move from one object to another, but it cannot be created or destroyed.

This is a fundamental conservation law, on the same level as conservation of energy and conservation of momentum. It holds in every known interaction: chemical reactions, nuclear decays, particle-antiparticle annihilation. If a process seems to "create" charge somewhere, an equal and opposite charge has appeared somewhere else.

Coulomb's law

Coulomb's law quantifies the electrostatic force between two point charges. It's the equation you'll use most in this unit.

Force between point charges

$F = k\frac{q_1 q_2}{r^2}$

• $F$ is the magnitude of the electrostatic force
• $q_1$ and $q_2$ are the two charges (in coulombs)
• $r$ is the distance between the charges (center to center)
• $k = 8.99 \times 10^{9} \; \text{N·m}^2/\text{C}^2$ is Coulomb's constant (in vacuum)

The force acts along the straight line connecting the two charges. If both charges have the same sign, the force is repulsive (pushes apart). If the signs differ, it's attractive (pulls together).

Notice the structural similarity to Newton's law of gravitation ($F = G\frac{m_1 m_2}{r^2}$). Both are inverse-square laws. The key difference: gravity is always attractive, while the electric force can attract or repel. At atomic scales, the electric force is enormously stronger than gravity.

Superposition principle

When more than two charges are present, the net force on any one charge equals the vector sum of the individual forces from every other charge. You calculate each pair separately using Coulomb's law, then add the force vectors.

Steps for a superposition problem:

1. Identify the charge you're finding the net force on (call it your "target" charge)
2. Calculate the force from each other charge on the target, one at a time, using Coulomb's law
3. Determine the direction of each force (attractive or repulsive)
4. Break each force into components if the charges aren't along a single line
5. Add the components to get the net force vector

This principle also applies to electric fields, not just forces.

Vector nature of force

Electrostatic force is a vector: it has both magnitude and direction. Getting the magnitude from Coulomb's law is only half the problem. You also need to figure out which way the force points.

• For two charges, the direction is along the line connecting them
• For systems with three or more charges, you'll need to resolve forces into $x$ and $y$ components and add them

This vector treatment is essential for problems involving charge equilibrium, electron deflection, and any arrangement where charges aren't all in a straight line.

Electric field

The electric field describes the influence a charge (or group of charges) exerts on the space around it. Instead of thinking about force between specific pairs of charges, the field tells you: if you placed a charge here, what force would it feel?

Definition and units

The electric field at a point is defined as the force per unit positive test charge:

$\vec{E} = \frac{\vec{F}}{q_0}$

where $q_0$ is a small positive test charge. The SI unit is newtons per coulomb (N/C), which is equivalent to volts per meter (V/m).

The electric field is a vector. Its direction is the direction of the force that a positive test charge would experience at that location.

Field lines representation

Field lines are a visual tool for mapping electric fields. Here are the rules for reading them:

• Field lines originate on positive charges and terminate on negative charges
• The direction of the field at any point is tangent to the field line passing through that point
• The strength of the field is indicated by how closely spaced the lines are (denser lines = stronger field)
• Field lines never cross each other (if they did, the field would point in two directions at one point, which is impossible)

Field due to a point charge

For a single point charge $q$, the electric field at distance $r$ is:

$\vec{E} = k\frac{q}{r^2}\hat{r}$

where $\hat{r}$ is the unit vector pointing radially outward from the charge.

• If $q$ is positive, the field points outward (away from the charge)
• If $q$ is negative, the field points inward (toward the charge)
• The field drops off as $1/r^2$, just like the force

For multiple point charges, you find the total field at any point by vector addition of the individual fields (superposition again).

Conductors vs. insulators

Materials are classified by how freely charge can move through them. This distinction determines how objects behave when placed in electric fields or given excess charge.

Charge distribution in conductors

Conductors (metals like copper, aluminum) have electrons that are free to move throughout the material. When a conductor is given excess charge or placed in an external field, those free electrons redistribute until they reach electrostatic equilibrium. At equilibrium:

• The net electric field inside the conductor is zero
• All excess charge sits on the outer surface
• Charge density is highest where the surface curves most sharply (small radius of curvature), which is why lightning rods are pointed

Electrostatic shielding

Because the field inside a conductor is zero at equilibrium, a hollow conductor blocks external electric fields from reaching its interior. This is the principle behind a Faraday cage.

Charges on the conductor's surface rearrange themselves to exactly cancel any external field inside. This is why you're relatively safe inside a car during a lightning storm, and why sensitive electronics are housed in metal enclosures.

Dielectric materials

Dielectrics are insulators (glass, rubber, plastic) that respond to electric fields through polarization. Their charges can't flow freely, but the bound charges shift slightly: positive ends of molecules tilt one way, negative ends the other.

• This polarization creates an internal field that partially opposes the applied field
• The net field inside the dielectric is reduced compared to vacuum
• Each dielectric is characterized by its dielectric constant ($\kappa$), which tells you by what factor it reduces the field
• Dielectrics are placed between capacitor plates to increase capacitance and energy storage

Charging methods

There are three main ways to give an object a net electric charge.

Friction and contact

When two different materials are rubbed together, electrons transfer from one surface to the other. This is the triboelectric effect. Which material gains electrons and which loses them depends on their relative electron affinities, ranked in the triboelectric series.

A classic example: rubbing a balloon on your hair transfers electrons to the balloon, leaving it negatively charged and your hair positively charged. Triboelectric charging can produce high voltages (thousands of volts) but very small currents.

Induction

Induction charges an object without direct contact. Here's how it works:

1. Bring a charged object (say, negative) near a neutral conductor
2. The negative charge repels free electrons in the conductor to the far side, leaving the near side positive
3. While the charged object is still nearby, connect the conductor to ground (touch it). Repelled electrons flow to ground.
4. Remove the ground connection, then remove the charged object
5. The conductor now has a net positive charge (opposite to the inducing charge)

Induction is used in devices like Van de Graaff generators and electrostatic precipitators.

Polarization

Polarization is what happens when an insulator is placed near a charged object. The bound charges in the insulator shift slightly, creating regions of local positive and negative charge on the surface. No net charge is transferred to the insulator; it remains neutral overall.

This is why a charged balloon sticks to a neutral wall. The balloon's charge polarizes the wall's surface, and the closer opposite charges create a net attractive force.

Measurement of charge

Several devices and experiments have been developed to detect and measure electric charge.

Electroscope

An electroscope detects the presence and sign of electric charge. In a gold leaf electroscope, two thin gold leaves hang from a conducting rod. When charge is transferred to the rod, both leaves acquire the same sign of charge and repel each other, spreading apart. The greater the charge, the wider the separation.

Electroscopes can also demonstrate charging by induction and were historically important in early electrostatics research.

Faraday cup

A Faraday cup is a metal container connected to a sensitive charge-measuring instrument (electrometer). When a beam of charged particles enters the cup, the cup captures them and the electrometer reads the total collected charge. Faraday cups are used in particle accelerators and mass spectrometers to measure ion beam currents with high precision.

Millikan oil drop experiment

Robert Millikan's 1909 experiment provided direct evidence for charge quantization. The setup:

1. Tiny oil droplets were sprayed into a chamber between two charged plates
2. The droplets picked up charge from ionized air
3. By adjusting the electric field between the plates, Millikan balanced the upward electric force against the downward gravitational force on individual droplets
4. From the balance condition, he calculated the charge on each droplet

Every measured charge turned out to be an integer multiple of $1.602 \times 10^{-19}$ C, confirming that charge comes in discrete packets. This experiment determined the value of the elementary charge $e$.

Applications of electrostatics

Van de Graaff generator

A Van de Graaff generator builds up large amounts of charge on a hollow metal dome. A motorized belt carries charge from a source at the base up to the dome, where it transfers to the dome's outer surface. Because charge always moves to the outside of a conductor, the belt can keep delivering more charge without limit (in principle).

These generators can reach voltages of several million volts. They're used in particle accelerators to give charged particles high kinetic energy, and they're a staple of physics demonstrations.

Electrostatic precipitators

Electrostatic precipitators remove fine particulates from exhaust gases in power plants and industrial facilities. The process works in two stages:

1. Exhaust gas passes through a region with high-voltage wires that ionize the air, giving the particulate matter a charge
2. The charged particles then pass between large collecting plates with the opposite charge, where Coulomb attraction pulls them out of the gas stream

These devices are highly efficient at capturing smoke, dust, and other fine particles, and they demonstrate charging by induction and Coulomb's law on an industrial scale.

Photocopiers and printers

Laser printers and photocopiers rely on electrostatics to transfer images onto paper:

1. A photoconductor drum is given a uniform electrostatic charge
2. A laser (or light from the original document) selectively discharges certain areas, creating a charge pattern that matches the desired image
3. Toner particles (fine charged powder) are attracted to the charged regions of the drum
4. The toner is transferred from the drum to the paper using an electric field
5. Heat and pressure fuse the toner permanently onto the paper

Charge in nature

Lightning and thunderstorms

Lightning is a massive natural electrostatic discharge. Inside a thundercloud, collisions between ice particles and water droplets separate charge: positive charge accumulates near the top of the cloud and negative charge near the bottom. When the charge difference becomes large enough, the air breaks down and a lightning bolt equalizes the charge between the cloud and the ground (or between different parts of the cloud).

Lightning channels can reach temperatures around 30,000 K, roughly five times the temperature of the Sun's surface.

Static electricity in daily life

You encounter electrostatic effects constantly:

• Static cling in clothes fresh from the dryer results from triboelectric charging as fabrics rub together
• Sparks when you touch a metal doorknob after walking across carpet happen because your body accumulated charge through friction, and it discharges rapidly through the small air gap
• Dust sticking to screens occurs because charged dust particles are attracted to polarized surfaces

These are all small-scale versions of the same physics behind lightning and industrial electrostatic devices.

Charge in biological systems

Electric charge is central to how living organisms function. Nerve impulses travel along neurons as waves of ion movement across cell membranes, driven by charge gradients. Cell membranes maintain a resting potential (about $-70 \; \text{mV}$ inside relative to outside) by actively pumping ions to sustain charge separation.

Some animals exploit bioelectricity more dramatically. Electric eels, for example, can generate pulses of over 600 V using stacks of specialized cells called electrocytes. Medical technologies like ECGs and defibrillators also depend on understanding how charge behaves in biological tissue.