🎢Principles of Physics II Unit 1 Review

Electric force describes how charged particles push or pull on each other. Coulomb's law gives you the math to calculate that force, and the superposition principle lets you handle situations with more than two charges. Together with the basic properties of charge, these ideas set the stage for everything else in electrostatics.

Coulomb's law

Coulomb's law tells you the magnitude of the electrostatic force between two point charges:

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

The force is directly proportional to the product of the two charges and inversely proportional to the square of the distance between them. Double the distance, and the force drops to one-quarter of its original value.

$k$ is Coulomb's constant: $8.99 \times 10^9 \, \text{N} \cdot \text{m}^2 / \text{C}^2$ (in vacuum)
$q_1, q_2$ are the magnitudes of the two charges (in coulombs)
$r$ is the center-to-center distance between the charges

The structure looks a lot like Newton's law of gravitation, but with charge replacing mass. One key difference: gravity is always attractive, while electric force can be attractive or repulsive depending on the signs of the charges. Like charges repel; opposite charges attract.

Point charges

A point charge is an idealized model where you treat a charged object as if all its charge sits at a single dimensionless point. This simplification works well when the objects are much smaller than the distance between them. For example, two charged spheres 1 meter apart that are each a few centimeters across can be treated as point charges with very little error.

This model lets you apply Coulomb's law directly and use vector algebra to find net forces when multiple charges are present.

Superposition principle

When more than two charges are involved, the net force on any one charge is the vector sum of the individual forces from every other charge:

$\vec{F}_\text{net} = \sum_{i=1}^n \vec{F}_i$

Each pair interaction is calculated independently using Coulomb's law, and then you add the resulting force vectors (not just magnitudes). This means you need to keep track of both direction and magnitude for each force.

To solve a superposition problem:

Identify the charge you're finding the net force on (the "target" charge).
Calculate the force from each other charge on the target using Coulomb's law.
Break each force into x- and y-components.
Sum all x-components and all y-components separately.
Combine the totals to find the magnitude and direction of the net force.

Properties of electric charge

Charge is a fundamental property of matter, just like mass. But unlike mass, charge comes in two varieties (positive and negative) and obeys strict rules about how it behaves.

Conservation of charge

The total electric charge in an isolated system never changes. Charges can move from one object to another, but they can't be created or destroyed. If you rub a balloon on your hair, electrons transfer from your hair to the balloon. Your hair becomes positive, the balloon becomes negative, but the total charge of the system stays the same.

This principle holds in every known physical process, from chemical reactions to nuclear decays.

Quantization of charge

Electric charge doesn't come in arbitrary amounts. It comes in discrete packets. The smallest unit is the magnitude of the charge on a single electron (or proton):

$e = 1.602 \times 10^{-19} \, \text{C}$

Every observable charge is an integer multiple of this value. You'll never measure a charge of $1.5e$ . This is what "quantized" means: charge only comes in whole-number multiples of a fundamental unit.

Conductors vs. insulators

Conductors (metals, salt solutions) have charge carriers that move freely through the material. Touch a charged conductor to a neutral one, and charge redistributes almost instantly.
Insulators (rubber, glass, plastic) hold charge in place. Charges stay where you put them.
Semiconductors (silicon, germanium) fall in between. Their conductivity can be tuned by adding impurities, a process called doping.

This distinction matters for every electrical device you'll encounter, from simple wires to transistors.

Electric field

Instead of thinking about forces between specific pairs of charges, the electric field gives you a way to describe what would happen to any charge placed at a given point in space.

Definition and concept

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

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

The field is a vector quantity. It has both magnitude and direction at every point in space. Once you know the field, you can find the force on any charge $q$ placed there: just multiply $\vec{F} = q\vec{E}$ .

Field lines

Field lines are a visual tool for mapping electric fields.

They point away from positive charges and toward negative charges.
Where lines are packed closely together, the field is strong. Where they spread apart, the field is weak.
Field lines never cross each other. If they did, that would imply two different field directions at the same point, which isn't physically possible.
For static (non-changing) electric fields, field lines don't form closed loops.

Field strength

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

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

The units are newtons per coulomb (N/C), which turn out to be equivalent to volts per meter (V/m). Just like force, the field weakens with the square of the distance from the source charge.

Coulomb's law, Coulomb's law - Wikipedia, the free encyclopedia

Forces on charged particles

Stationary charges

A stationary charge sitting in an electric field experiences a force $\vec{F} = q\vec{E}$ . If multiple charges are nearby, you find the net force using superposition. A charge is in equilibrium when the net force on it is zero.

On a conductor, free charges redistribute themselves until the electric field inside the conductor is zero. This redistribution explains why excess charge on a conductor always ends up on its surface.

Moving charges

Once a charge starts moving, things get more interesting. A moving charge in combined electric and magnetic fields experiences the Lorentz force:

$\vec{F} = q(\vec{E} + \vec{v} \times \vec{B})$

The electric part ( $q\vec{E}$ ) pushes the charge along the field direction regardless of how it's moving. The magnetic part ( $q\vec{v} \times \vec{B}$ ) acts perpendicular to both the velocity and the magnetic field, curving the charge's path without changing its speed.

This is the principle behind particle accelerators and mass spectrometers.

Multiple charge interactions

With three or more charges, you apply superposition: calculate the force from each pair and add the vectors. These calculations require careful bookkeeping of directions.

Multiple charge systems can produce stable arrangements (like ions locked in a crystal lattice) or dynamic ones (like charged particles in a plasma). The math gets complex quickly, which is why symmetric arrangements and simplifying assumptions are so valuable.

Electric dipoles

An electric dipole is a pair of equal and opposite charges ( $+q$ and $-q$ ) separated by a small distance $d$ . Dipoles show up everywhere, from water molecules to radio antennas.

Dipole moment

The dipole moment is a vector that captures the dipole's strength and orientation:

$\vec{p} = q\vec{d}$

Here, $\vec{d}$ points from the negative charge to the positive charge. The units are coulomb-meters (C·m). A larger charge or a larger separation means a stronger dipole moment. Water, for instance, is a polar molecule with a permanent dipole moment, which is why it's such a good solvent.

Torque on dipoles

Place a dipole in a uniform electric field and it experiences a torque that tries to rotate it into alignment with the field:

$\vec{\tau} = \vec{p} \times \vec{E}$

The torque is maximum when the dipole is perpendicular to the field and zero when it's aligned (or anti-aligned). This is why polar molecules tend to orient themselves in an applied field.

Dipole in electric field

In a uniform field, a dipole feels torque but no net force (the forces on the two charges are equal and opposite). In a non-uniform field, the dipole also experiences a net force because the field is stronger on one charge than the other.

The potential energy of a dipole in a field is $U = -\vec{p} \cdot \vec{E}$ . Energy is minimized when the dipole is aligned with the field, which is the stable orientation.

Applications of electric force

Electrostatic precipitators

These devices clean exhaust gases in power plants and factories. The process works in stages:

Dirty gas flows past a set of wires carrying a high voltage, which creates a corona discharge.
The corona discharge ionizes nearby gas molecules, and these ions attach to particulate matter, giving the particles a charge.
Charged particles drift toward oppositely charged collection plates.
Particles accumulate on the plates and are periodically shaken off for disposal.

Efficiency depends on particle size, the strength of the electric field, and how long the gas spends in the device.

Photocopiers and printers

Laser printers and photocopiers rely on electrostatics to form images:

A photoconductor drum is given a uniform electrostatic charge.
A laser (or bright light reflecting off the original document) selectively discharges areas of the drum, creating a charge pattern that matches the image.
Toner particles, which carry an opposite charge, are attracted to the charged areas of the drum.
The toner is transferred from the drum onto paper using an electric field.
Heat and pressure fuse the toner permanently to the paper.

Van de Graaff generator

A Van de Graaff generator builds up large amounts of charge on a hollow metal sphere:

A motor-driven belt picks up charge from a source (often via friction or a small corona discharge).
The belt carries charge upward into the hollow sphere.
Charge transfers to the sphere's outer surface, where it accumulates.
The sphere reaches very high voltages (hundreds of thousands of volts in lab models).

The strong electric field near the sphere can ionize air, producing visible sparks. These generators are used as particle accelerators in research and as classroom demonstration tools.

Experimental methods

Millikan oil drop experiment

Robert Millikan's 1909 experiment proved that charge is quantized and measured the electron's charge.

Tiny oil droplets were sprayed into a chamber and allowed to pick up charge (from friction or ionizing radiation).
An electric field between two parallel plates was adjusted until the upward electric force on a droplet exactly balanced the downward gravitational force.
By measuring the voltage needed to suspend each droplet and knowing the droplet's mass, Millikan calculated the charge on each one.
Every measured charge turned out to be an integer multiple of $1.602 \times 10^{-19} \, \text{C}$ .

Electroscope

An electroscope is a simple device for detecting charge. It consists of a metal rod with two thin metal leaves (often gold foil) hanging from the bottom. 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 leaves separate.

Electroscopes can also demonstrate charging by induction: bring a charged object near (but not touching) the top of the electroscope, and the leaves still deflect because charges within the electroscope redistribute.

Faraday cage

A Faraday cage is a hollow conductor (a metal box or mesh) that shields its interior from external electric fields. When an external field is applied, free charges on the cage rearrange themselves to cancel the field inside. The result: the interior has zero electric field regardless of what's happening outside.

Practical uses include shielding sensitive electronics from electromagnetic interference, the metal body of your car protecting you during a lightning strike, and the mesh screen in a microwave oven door keeping radiation inside.

Electric force in materials

Dielectrics

A dielectric is an insulating material that becomes polarized when placed in an electric field. The molecules inside shift slightly (or rotate, if they're polar), creating an internal field that partially opposes the applied field. This reduces the net field inside the material.

Dielectrics are characterized by their dielectric constant ( $\varepsilon_r$ , also called relative permittivity). A higher dielectric constant means more polarization and a greater reduction in field strength. Common dielectrics include ceramics, plastics, glass, and even air.

Polarization

Polarization is the process by which dipoles within a material align with an applied electric field. Two mechanisms can cause it:

Orientation polarization: Molecules with permanent dipole moments (like water) rotate to align with the field.
Induced polarization: In non-polar molecules, the applied field distorts the electron cloud, creating a temporary dipole.

Both effects produce a net dipole moment in the material, which modifies the electric field inside and around it.

Permittivity

Permittivity ( $\varepsilon$ ) measures how easily a material stores electric energy in an electric field. It's defined as:

$\varepsilon = \varepsilon_r \varepsilon_0$

where $\varepsilon_0 = 8.85 \times 10^{-12} \, \text{C}^2/(\text{N} \cdot \text{m}^2)$ is the permittivity of free space and $\varepsilon_r$ is the relative permittivity of the material. Inserting a dielectric with $\varepsilon_r = 3$ into a capacitor triples its capacitance compared to vacuum.

Permittivity also affects how electromagnetic waves propagate through a material. In some materials, $\varepsilon_r$ varies with the frequency of the applied field, a phenomenon called dielectric dispersion.

Limitations and approximations

Point charge approximation

Treating objects as point charges works well when the objects are small compared to the distance between them. It breaks down when charges are close together or when the charge is spread over a large region. In those cases, you need to account for the actual charge distribution, sometimes using techniques like multipole expansion.

Infinite plane approximation

A large, flat charged surface can be modeled as an infinite plane. This produces a uniform electric field perpendicular to the surface, which greatly simplifies calculations. The approximation holds as long as you're looking at points much closer to the surface than to its edges. It's commonly used when analyzing parallel-plate capacitors.

Gauss's law connection

Gauss's law relates the total electric flux through a closed surface to the charge enclosed:

$\oint \vec{E} \cdot d\vec{A} = \frac{Q_\text{enc}}{\varepsilon_0}$

For charge distributions with high symmetry (spherical, cylindrical, or planar), Gauss's law provides a much faster way to find the electric field than summing up Coulomb's law contributions from every bit of charge. It doesn't give you new physics beyond Coulomb's law, but it gives you a powerful shortcut when the geometry cooperates.

🎢Principles of Physics II Unit 1 Review