2.1 Electric field

Definition of electric field

An electric field describes the influence a charge exerts on the space around it. Any other charge placed in that space will feel a force, and the electric field tells you exactly how strong that force would be and in which direction it points. This concept is central to electromagnetism and shows up everywhere, from circuit design to understanding how molecules interact.

Electric fields are vector quantities, meaning they have both magnitude and direction. They're measured in newtons per coulomb (N/C) or equivalently volts per meter (V/m).

Electric field vs electric force

The electric field at a point in space tells you the force per unit charge that a positive test charge would experience there. The key distinction: the electric field exists whether or not a test charge is present, while the electric force is what a specific charge actually feels when placed in that field.

• Electric field ($\vec{E}$): force per unit charge at a location, independent of whatever charge you place there
• Electric force ($\vec{F}$): the actual force on a particular charge, given by $\vec{F} = q\vec{E}$
• A stronger charge feels a stronger force in the same field, but the field itself doesn't change

The direction of the electric field is defined as the direction a positive test charge would be pushed.

Vector nature of electric field

At every point in space, the electric field has both a magnitude (how strong) and a direction (which way). This matters because when multiple charges create overlapping fields, you add them as vectors, not just numbers.

• Field vectors can be drawn as arrows: longer arrows mean stronger fields
• When combining fields from multiple sources, you must account for direction (vector addition), not just add magnitudes
• The scalar quantity electric potential is related to the electric field through a gradient operation: $\vec{E} = -\nabla V$

Electric field of point charges

Point charges are the simplest case and the building block for everything else. Once you understand how a single point charge creates a field, you can build up to more complex charge distributions by adding contributions together.

Coulomb's law for point charges

The electric field created by a single point charge $q$ at a distance $r$ is:

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

where $k = 8.99 \times 10^9 \, \text{N·m}^2/\text{C}^2$ is Coulomb's constant.

• For a positive charge, the field points radially outward
• For a negative charge, the field points radially inward (toward the charge)
• The inverse-square relationship means doubling your distance from the charge cuts the field strength to one-quarter

Superposition principle

When multiple charges are present, the total electric field at any point is the vector sum of the individual fields from each charge:

$\vec{E}_{\text{total}} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \ldots$

This is powerful because it lets you handle complex arrangements by breaking them into simpler pieces. Calculate the field from each charge separately, then add the vectors. This principle applies to both discrete point charges and continuous charge distributions.

Electric field lines

Field lines are a visual tool for representing electric fields. They don't physically exist, but they give you an intuitive picture of what the field looks like around a charge or group of charges.

Properties of field lines

• They start on positive charges and end on negative charges (or extend to infinity if there's no nearby opposite charge)
• They never cross. Since the field has a unique direction at every point, two lines crossing would imply two directions at the same spot, which is impossible.
• Closer spacing between lines means a stronger field
• The tangent to a field line at any point gives the direction of $\vec{E}$ at that point

Field line patterns

• Point charge: radial lines spreading outward (positive) or converging inward (negative)
• Uniform field: parallel, evenly spaced lines, indicating constant magnitude and direction
• Dipole: lines curve from the positive charge to the negative charge, with some lines extending to infinity at the edges
• Quadrupole and more complex arrangements create patterns with regions of stronger and weaker fields

Electric field of continuous charge distributions

Real objects aren't point charges. A charged rod, a metal plate, or a charged sphere all have charge spread over a region. To find the electric field, you treat the object as made of infinitely many tiny charge elements $dq$, calculate each one's contribution using Coulomb's law, and integrate.

Symmetry is your best friend here. If the charge distribution has a nice geometric symmetry, many components of the field cancel out, and the integral becomes much simpler.

Line charges

Charge distributed along a one-dimensional path (a wire, rod, or ring). You integrate contributions from small segments $dq$ along the line.

For a long straight wire with uniform linear charge density $\lambda$, the field at a perpendicular distance $r$ from the wire falls off as $1/r$ (not $1/r^2$ like a point charge), because the charge extends in one dimension.

Surface charges

Charge spread over a two-dimensional surface, such as a flat plate, sphere, or cylinder.

• For an infinite uniformly charged plane with surface charge density $\sigma$, the field is $E = \frac{\sigma}{2\epsilon_0}$, directed perpendicular to the surface. This field is constant regardless of distance, which is a unique and useful result.
• Gauss's law is especially helpful for these high-symmetry cases.

Volume charges

Charge distributed throughout a three-dimensional region (a solid sphere, cylinder, etc.). You integrate over volume elements $dV$, each contributing $dq = \rho \, dV$ where $\rho$ is the volume charge density.

For a uniformly charged solid sphere, the field outside behaves like a point charge, while inside the sphere the field increases linearly with distance from the center.

Dipoles in electric fields

A dipole is a pair of equal and opposite charges ($+q$ and $-q$) separated by a small distance. Dipoles are everywhere in nature: water molecules, for instance, are permanent dipoles. Understanding how dipoles behave in external fields is key to understanding dielectric materials and molecular physics.

Electric dipole moment

The dipole moment is a vector that captures both the strength and orientation of a dipole:

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

where $q$ is the magnitude of either charge and $\vec{d}$ is the displacement vector pointing from the negative charge to the positive charge. It's measured in coulomb-meters (C·m). The dipole moment determines how the dipole responds to external fields.

Torque on electric dipoles

An external electric field exerts a torque on a dipole, trying to rotate it into alignment with the field:

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

• Torque is maximum when the dipole is perpendicular to the field ($\tau = pE$)
• Torque is zero when the dipole is aligned with or directly against the field

This is why polar molecules tend to orient themselves in an applied electric field, which is the basis of dielectric polarization.

Gauss's law

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

$\oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\epsilon_0}$

It's mathematically equivalent to Coulomb's law, but for charge distributions with high symmetry (spherical, cylindrical, or planar), it makes calculating the electric field far easier.

Gaussian surfaces

A Gaussian surface is an imaginary closed surface you choose to make the math simple. It has no physical reality; you pick it based on the symmetry of the problem.

• Spherical Gaussian surface for point charges or spherical distributions
• Cylindrical Gaussian surface for long wires or cylindrical distributions
• Rectangular box (pillbox) for infinite planes

The goal is to choose a surface where $\vec{E}$ is either constant and parallel to $d\vec{A}$, or perpendicular to $d\vec{A}$ (contributing zero flux), so the integral simplifies.

Applications of Gauss's law

• Deriving the electric field of spheres, cylinders, and infinite planes
• Proving that the electric field inside a conductor in electrostatic equilibrium is zero
• Showing that excess charge on a conductor sits entirely on its surface
• Analyzing the field inside and outside capacitors and dielectric materials

Electric field in conductors

Conductors have mobile charge carriers (usually free electrons) that can move in response to electric fields. This mobility leads to some important and sometimes surprising results.

Electrostatic equilibrium

When a conductor reaches electrostatic equilibrium (charges stop moving), three things are true:

1. The electric field inside the conductor is zero
2. Any excess charge resides entirely on the surface
3. The surface is an equipotential (same voltage everywhere on it)

If the internal field weren't zero, free charges would still be moving, which contradicts equilibrium.

Shielding effect

Because charges on a conductor's surface rearrange to cancel any internal field, the interior is completely shielded from external electric fields. This is the principle behind a Faraday cage.

• A hollow conducting shell blocks external fields from reaching the inside
• This is why sensitive electronics are housed in metal enclosures
• The shielding works regardless of the shape of the conductor, as long as it fully encloses the region

Electric field in dielectrics

Dielectrics are insulating materials (glass, rubber, water, etc.) that don't conduct electricity but respond to electric fields through polarization. They're essential in capacitor design and electrical insulation.

Polarization of dielectrics

When you place a dielectric in an external electric field, its molecules respond in one of two ways:

• Polar molecules (like water) physically rotate to align their existing dipole moments with the field
• Nonpolar molecules develop induced dipole moments as the field slightly shifts the electron cloud relative to the nucleus

In both cases, bound charges appear on the dielectric's surfaces. These bound charges create a field that opposes the external field, reducing the net field inside the material.

Dielectric constant

The dielectric constant $\kappa$ (kappa) quantifies how much a material reduces the electric field compared to vacuum:

• Defined as $\kappa = \frac{\epsilon}{\epsilon_0}$, where $\epsilon$ is the material's permittivity and $\epsilon_0$ is the permittivity of free space
• It's dimensionless and always $\geq 1$ (vacuum has $\kappa = 1$)
• Higher $\kappa$ means the material reduces the field more and increases capacitance
• For example, water has $\kappa \approx 80$, which is why it's such an effective dielectric

Energy and potential in electric fields

Electric fields store energy and create potential differences between points. These concepts connect force and field to the work and energy ideas you already know from mechanics.

Electric potential energy

The potential energy of a charge configuration depends on the positions of the charges relative to each other.

• For a charge $q$ at a point where the electric potential is $V$: $U = qV$
• For a dipole in an external field: $U = -\vec{p} \cdot \vec{E}$
  • Energy is lowest (most stable) when the dipole is aligned with the field
  • Energy is highest when the dipole points opposite to the field
• Changes in potential energy equal the work done by electric forces as charges move

Electric potential vs electric field

Electric potential ($V$) is a scalar measured in volts (J/C). It represents the work per unit charge needed to bring a test charge from infinity to that point.

The electric field is the negative gradient of the potential:

$\vec{E} = -\nabla V$

This means the field points in the direction of decreasing potential, and its magnitude equals the rate at which potential changes with distance.

• Equipotential surfaces are always perpendicular to electric field lines
• No work is done moving a charge along an equipotential surface
• Closely spaced equipotential lines indicate a strong electric field

Experimental methods

Several landmark experiments shaped our understanding of electric charge and fields. Two of the most important are worth knowing in detail.

Millikan oil drop experiment

This experiment determined the fundamental unit of electric charge (the charge of a single electron, $e = 1.6 \times 10^{-19}$ C).

1. Tiny oil droplets are sprayed between two horizontal charged plates
2. The electric field between the plates is adjusted until the upward electric force on a droplet exactly balances the downward gravitational force
3. At balance: $qE = mg$, so the charge on the droplet can be calculated
4. Repeating with many droplets, Millikan found that all measured charges were integer multiples of $e$

This demonstrated that electric charge is quantized, not continuous.

Cathode ray tube experiments

J.J. Thomson used cathode ray tubes to discover the electron in 1897.

• He showed that cathode rays were streams of negatively charged particles (not waves)
• By deflecting the beam with both electric and magnetic fields, he measured the charge-to-mass ratio ($e/m$) of the electron
• The same ratio appeared regardless of the cathode material, proving the electron is a universal constituent of matter

Applications of electric fields

Particle accelerators

Particle accelerators use strong electric fields to accelerate charged particles to high speeds.

• Linear accelerators pass particles through a series of electrodes with alternating potentials, giving them a "kick" at each gap
• Cyclotrons and synchrotrons combine electric fields (for acceleration) with magnetic fields (for circular steering)
• Applications range from fundamental particle physics research to medical radiation therapy and materials analysis

Electrostatic precipitators

These devices clean exhaust gases by using electric fields to remove particulate matter.

1. Gas containing particles flows past a charging electrode, which ionizes the particles
2. The charged particles are then attracted to grounded collection plates by the electric field
3. Accumulated particles are periodically removed from the plates

They're widely used in coal-fired power plants, cement factories, and other industrial settings to reduce air pollution by capturing fine dust, smoke, and aerosols.