Charge can be positive or negative. An object is neutral when its net charge is zero. Like charges repel, and opposite charges attract. Charge is quantized and measured in coulombs ©.

In conductors (such as metals), charges can move freely, which makes it easy for charge - and therefore electrical energy - to be transferred.

In insulators, charges don’t move easily. That’s why nonmetals are typically poor conductors.

A key principle is charge conservation: charge can’t be created or destroyed. It can only be transferred from one object to another.

Coulomb’s law

Coulomb’s law describes the electrostatic force between two charged particles. The force is directly proportional to the product of the charges ( $q_{1}$ and $q_{2}$ ) and inversely proportional to the square of the distance ( $r$ ) between them:

$F = \frac{k q _{1} q _{2}}{r ^{2}}$

where $k$ is Coulomb’s constant (approximately $9 \times 1 0^{9} N \cdot m^{2} / C^{2}$ ).

If the charges have the same sign, the force is repulsive.
If the charges have opposite signs, the force is attractive.

Electric field e

The electric field is a vector quantity that tells you the force per unit charge at a point in space. Its units are newtons per coulomb ( $N/C$ ).

A common way to visualize an electric field is with field lines, which show the direction and relative strength of the field:

Closely spaced lines signify a stronger field.
More widely spaced lines indicate a weaker field.
For a single positive charge, lines radiate outward; for a negative charge, they converge inward.

Electric field lines around a positive charge

Electric field lines around point charges

Adding electrical charges

Adding electric fields from two point charges

The electric fields $E_{1}$ and $E_{2}$ at the origin $O$ add to $E_{tot}$ .

The electric field strength at the origin due to $q_{1}$ is labeled $E_{1}$ and is calculated as:

$E_{1} = k \frac{q _{1}}{r _{1}^{2}} = (8.99 \times 1 0^{9} N \cdot m^{2} / C^{2}) \cdot \frac{5.00 \times 1 0 ^{- 9} C}{( 2.00 \times 1 0 ^{- 2} m ) ^{2}}$

$E_{1} = 1.124 \times 1 0^{5} N/C$

Similarly, $E_{2}$ is:

$E_{2} = k \frac{q _{2}}{r _{2}^{2}} = (8.99 \times 1 0^{9} N \cdot m^{2} / C^{2}) \cdot \frac{10.0 \times 1 0 ^{- 9} C}{( 4.00 \times 1 0 ^{- 2} m ) ^{2}}$

$E_{2} = 0.5619 \times 1 0^{5} N/C$

The magnitude of the total field $E_{tot}$ is:

$E_{tot} = E_{1}^{2} + E_{2}^{2} = (1.124 \times 1 0^{5})^{2} + (0.5619 \times 1 0^{5})^{2} = 1.26 \times 1 0^{5} N/C$

The direction is:

$θ = tan^{- 1} (\frac{E _{1}}{E _{2}}) = tan^{- 1} (\frac{1.124 \times 1 0 ^{5}}{0.5619 \times 1 0 ^{5}}) = 63. 4^{\circ}$

or $63. 4^{\circ}$ above the x-axis.

In a dipole system, field lines emerge from the positive end and enter the negative end. For two like charges, the lines repel one another. The overall pattern reverses for negative charges compared to positive charges.

Electric fields of like and unlike charges

When multiple charges are present, the net electric field at any location is found by taking the vector sum of the individual fields.

In specific configurations:

In a capacitor, the electric field between the plates is typically uniform except near the edges, where fringe effects occur.
Around a long, charged wire or a cylinder, the electric field extends radially outward.
For a conducting cylinder, the electric field inside is zero.

Absolute potential and potential difference

The absolute potential at a point is the energy per unit charge that a test charge would have due to a source charge.

It can be written as $V = U / q_{0} = k q / r$ , where:

$U$ is the electric potential energy of the test charge $q_{0}$
$q$ is the charge creating the potential
$r$ is the distance from that charge

When multiple charges are present, the total potential is the algebraic sum of the individual potentials (positive charges create positive potentials, and negative charges create negative potentials).

The unit of potential is the volt (V), which is equivalent to joules per coulomb (J/C).

The potential difference ( $Δ V$ ) is the difference between the potentials at two points:

$Δ V = V_{B} - V_{A}$ .

This quantity is used in many applications, such as the voltage across the plates of a capacitor or between the positive and negative terminals of a battery.

Equipotential lines and electrical dipoles

Equipotential lines are contours along which the potential remains constant. Because there is no change in potential when moving along these lines, no work is done. Equipotential lines are always perpendicular to electric field lines.

An electric dipole consists of a positive and a negative charge separated by a distance. In an electric field, a dipole aligns so that its positive end points in the direction of the field. To find the potential created by a dipole at a point, calculate the potential from each charge and then add them.

Electrostatic energy, electric potential at a point in space

Electrostatic induction happens when a charged object causes charges in a nearby neutral object to redistribute, without any direct conduction of charge between them. The charged object’s electric field polarizes the neutral object by shifting its internal charge distribution.

A common example is a sweater rubbed against hair causing hair to stand up: the charged sweater creates an electric field that rearranges charges in the hair, producing a noticeable effect even though no electrons are physically transferred. This process is static, meaning the induced polarization occurs due to the presence of the charged species.

A point charge is an idealized model where an electric charge is treated as if it is concentrated at a single point in space, even though real charges occupy a small but finite volume. This assumption simplifies many electric field and potential calculations.

Gauss’s law states that the net electric flux ( $Φ_{E}$ ) through any closed surface equals the total charge ( $q$ ) enclosed divided by the permittivity of free space ( $ϵ_{0}$ ). Mathematically:

$Φ_{E} = q / ϵ_{0}$ .

The electric flux is calculated as the product of the electric field ( $E$ ) and the area ( $A$ ) it penetrates, multiplied by the cosine of the angle ( $θ$ ) between the field direction and the normal to the surface:

$Φ_{E} = E A cos θ$ .

This law implies that if no charge is contained within a closed surface, the net electric flux through that surface is zero.

The Faraday cage is a practical application of Gauss’s law. It is a closed conducting shell where the internal electric field is nullified by the rearrangement of surface charges that cancel external fields. In popular culture, it’s often described as a way to protect electronics from an EMP event.

Charge, conductors, charge conservation, and insulators

Charge: positive or negative, quantized, measured in coulombs ©
Conductors: charges move freely; insulators: charges do not move easily
Charge conservation: total charge cannot be created or destroyed, only transferred

Coulomb’s law

Electrostatic force: $F = \frac{k q _{1} q _{2}}{r ^{2}}$ , $k \approx 9 \times 1 0^{9} N \cdot m^{2} / C^{2}$
Force is repulsive for like charges, attractive for opposite charges

Electric field E

Electric field ( $E$ ): force per unit charge, units N/C
Field lines: direction shows force on positive charge, density indicates strength
- Outward from positive, inward to negative

Adding electrical charges

Net electric field: vector sum of individual fields
Example: $E_{tot} = E_{1}^{2} + E_{2}^{2}$ , direction via $θ = tan^{- 1} (E_{1} / E_{2})$
Special cases:
- Dipole: field lines from positive to negative
- Capacitor: uniform field between plates (except edges)
- Cylinder/wire: radial field; inside conductor, field is zero

Absolute potential and potential difference

Absolute potential: $V = k q / r$ , units volts (V = J/C)
Potential difference: $Δ V = V_{B} - V_{A}$
Total potential: algebraic sum of potentials from all charges

Equipotential lines and electrical dipoles

Equipotential lines: constant potential, perpendicular to electric field lines, no work done moving along them
Electric dipole: two opposite charges separated by distance, aligns with field, total potential is sum from each charge

Electrostatic energy, electric potential at a point in space

Electrostatic induction: redistribution of charges in a neutral object by a nearby charged object, no direct contact
Point charge: idealized charge at a single point, simplifies calculations
Gauss’s law: $Φ_{E} = q / ϵ_{0}$ , electric flux through closed surface equals enclosed charge over permittivity
- $Φ_{E} = E A cos θ$
- No enclosed charge $\to$ net flux is zero
Faraday cage: closed conductor cancels internal electric fields, practical application of Gauss’s law

Electrostatics