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Electrostatics
Charges, fields, potentials, capacitors and dielectrics. 54 topics, from beginner to advanced. Open the interactive version with labs and quizzes
Electrostatics studies electric charge at rest. This area covers the two kinds of charge, conservation and quantisation of charge, Coulomb's law, electric fields and potential, Gauss's law in simple geometries, capacitors and dielectrics. These ideas explain everything from static cling to the design of electronic components.
How the topics connect
Electrostatics starts with electric charge, the force between charges and the electric field. Potential, energy, capacitance and Gauss's law build the tools for analysing static charges, and they lead naturally to the currents and magnetism that follow in later areas.
Real-world uses
Choose capacitor sizes and voltages for smoothing circuits, timing circuits and energy storage in flash units.
Use electrostatic precipitators to remove dust from smokestacks and electrostatic spray painting to coat car bodies evenly.
Understand how charged drums attract toner particles to form an image on paper.
Explain how electric fields build up in storm clouds and why a car or metal cage protects occupants from lightning.
Engineers use capacitors to store energy in camera flashes and to smooth power supplies in computers.
Electrostatic precipitators clean factory exhaust, while engineers use shielding and earthing to protect equipment from static.
Electric charge
A property of matter that creates and feels electric force.
Charge is a fundamental property of matter, positive or negative, which produces and feels electric forces. Like charges repel, unlike attract. The SI unit is the coulomb.
Rubbing a balloon on hair transfers electrons.
Charge is carried by particles, mostly electrons and protons.
Conductors and insulators
Charges move freely in conductors but not in insulators.
In conductors (metals) outer electrons move freely; insulators (glass, plastic) hold them tightly. Semiconductors lie between.
Copper wire conducts, rubber coating insulates.
Even insulators can be charged but charges stay in place.
Charging by friction, contact, induction
Three ways to charge objects.
Friction transfers electrons between materials; contact shares charge with a touching body; induction charges a conductor without contact by bringing a charged object near and earthing.
Gold-leaf electroscope charged by induction.
Induction produces opposite charge on the nearest side.
Conservation of charge
Total charge in an isolated system is constant.
The total charge in an isolated system never changes: charge is neither created nor destroyed, only transferred. It holds even in particle reactions.
In pair production a photon gives +e and −e, total zero.
Net charge, not the number of particles, is conserved.
Quantisation of charge
Charge comes in multiples of e.
q = ne, e=1.6×10⁻¹⁹ CFree charge occurs only in integer multiples of e = 1.602×10⁻¹⁹ C (quarks carry e/3 but are not seen alone). Millikan's experiment showed this.
1 C is about 6.24×10¹⁸ electrons.
At everyday scales it looks continuous.
Coulomb's law
Force between point charges.
F = kq₁q₂/r²The force between two point charges is F = kq₁q₂/r², along the line joining them, repulsive if like, attractive if unlike.
Two 1 μC charges 10 cm apart: 0.9 N.
Use the magnitudes for the size and signs for direction.
Coulomb constant
k = 1/4πε₀.
k = 9×10⁹ N·m²/C²k = 1/(4πε₀) = 8.99×10⁹ N·m²/C²; ε₀ = 8.85×10⁻¹² C²/N·m² is the permittivity of free space. In a medium ε₀ is replaced by ε = κε₀.
Water's κ = 80 reduces the force 80-fold.
The huge size of k shows how strong electrostatics is.
Superposition of forces
Net force is the vector sum from each charge.
The force on a charge due to several others is the vector sum of the individual Coulomb forces. Each pair interacts independently.
A charge at the centre of a symmetric ring of equal charges feels zero net force.
Add components, not magnitudes.
Electric field
Force per unit positive charge.
E = F/qThe field E = F/q is the force per unit positive test charge. It exists in space around charges and acts on any charge placed there: F = qE.
A 10⁻⁶ C charge in a 5000 N/C field feels 5 mN.
The test charge must be small to avoid disturbing the field.
Field of a point charge
Field decreases as inverse square.
E = kq/r²E = kq/r², directed away from a positive charge and toward a negative one. It follows an inverse-square law.
1 nC at 1 m: 9 N/C.
Superpose vectorially for multiple charges.
Electric field lines
Lines begin on positive and end on negative charges.
efieldLines start on positive charges and end on negative ones, tangent to the field direction, and denser lines mean a stronger field. They never cross.
A dipole shows curved lines between the charges.
Field lines are a visual device, not physical objects.
Electric dipole
Two equal and opposite charges separated by d.
p = qdTwo equal and opposite charges ±q separated by d form a dipole with moment p = qd (from − to +). Water molecules are dipoles.
p = 1.6×10⁻¹⁹ × 10⁻¹⁰ = 1.6×10⁻²⁹ C·m.
Net charge is zero but there's still a field.
Field of a dipole
On the axis the field is 2kp/r³.
E = 2kp/r³On the axis far away E = 2kp/r³; on the equatorial line E = kp/r³ with opposite direction. It falls as 1/r³, faster than a point charge.
Biological molecules interact through dipole fields.
Valid for r ≫ d.
Torque on a dipole in a field
A dipole rotates to align with the field.
τ = pE sinθA dipole in a uniform field feels torque τ = p × E, magnitude pE sinθ, rotating it to align with E. There's no net force in a uniform field.
p = 10⁻²⁹ C·m in 10⁵ N/C at 90°: 10⁻²⁴ N·m.
In a non-uniform field there's also a net force.
Potential energy of a dipole
Lowest when aligned.
U = −p·EU = −p·E = −pE cosθ. It's lowest when aligned (θ = 0) and highest when opposite. Microwaves heat food by rotating water dipoles.
Work to rotate by 180°: 2pE.
Zero reference is at θ = 90°.
Electric flux
Field lines through a surface.
Φ = ∮E·dAFlux Φ = ∮E·dA counts the field lines passing through a surface. For a uniform field through a flat area, Φ = EA cosθ.
1000 N/C through 0.5 m² face-on: 500 N·m²/C.
Flux is a scalar and can be negative.
Gauss's law
Flux through a closed surface equals enclosed charge over ε₀.
∮E·dA = q/ε₀The net flux through any closed surface equals the enclosed charge divided by ε₀: ∮E·dA = Q_enc/ε₀. It makes fields of symmetrical charge distributions easy.
A sphere with 1 nC enclosed: Φ = 113 N·m²/C.
Only enclosed charges count, but all charges produce the field.
Field of infinite line charge
Falls as 1/r.
E = λ/2πε₀rBy Gauss with a cylinder: E = λ/(2πε₀r), pointing radially, falling as 1/r.
λ = 1 nC/m at 0.5 m: 36 N/C.
Valid for lines much longer than the distance.
Field of infinite plane sheet
Uniform and independent of distance.
E = σ/2ε₀E = σ/2ε₀ on each side, uniform and independent of distance. Two oppositely charged parallel plates give E = σ/ε₀ between them and zero outside.
σ = 1 μC/m²: 56,500 N/C.
Valid near the middle of large sheets.
Field of a charged shell
Zero inside, point-like outside.
Outside a uniformly charged spherical shell the field equals that of a point charge at the centre; inside it is zero. The same holds for gravity.
Charge on a hollow metal sphere sits on the outside.
Not the same for non-spherical shapes.
Field of a uniformly charged sphere
Grows linearly inside.
E = kQr/R³Outside E = kQ/r²; inside a uniformly charged insulating sphere E = kQr/R³, growing linearly from the centre.
At r = R/2 the field is half the surface value.
A conducting sphere has zero field inside.
Field of a conductor
Inside a conductor E is zero; at the surface E = σ/ε₀.
In electrostatic equilibrium the field inside a conductor is zero, charge resides on the surface, and the field just outside is perpendicular to the surface with E = σ/ε₀.
Charges concentrate at sharp points (lightning rods).
These conditions assume no current flowing.
Electrostatic shielding
A cavity inside a conductor is field free.
A hollow conductor screens its interior from external fields (Faraday cage). Cars and aircraft protect passengers from lightning in this way.
Mobile signal drops inside a lift.
A cavity with a charge inside has induced surface charge.
Electric potential
Potential energy per unit charge.
V = kq/rThe potential V = U/q is the work per unit charge to bring a test charge from infinity. For a point charge V = kq/r. It's a scalar and easy to add.
1 μC at 0.3 m: 30 kV.
Zero reference can be chosen, usually at infinity.
Potential difference
Work per unit charge to move between points.
ΔV = −∫E·dlΔV = −∫E·dl is the work per unit charge to move between two points; it's the voltage. A charge q gains energy qΔV.
An electron accelerated through 100 V gains 100 eV.
The potential difference is independent of the path.
Equipotential surfaces
Surfaces of constant potential are perpendicular to field lines.
Surfaces on which V is constant. They are perpendicular to field lines, and no work is done moving a charge along them. A conductor's surface is an equipotential.
Topographic contours are like gravitational equipotentials.
Closer spacing means a stronger field.
Relation of E and V
Field is the negative gradient of potential.
E = −dV/dxThe field is the negative gradient of potential, E = −∇V. In 1D E = −dV/dx; for a uniform field V = Ed.
A 12 V across 2 mm gap: 6000 V/m.
E points toward decreasing V.
Electron volt
Energy gained by an electron across one volt.
1 eV = 1.6×10⁻¹⁹ J1 eV is the energy gained by an electron moving through 1 V: 1.602×10⁻¹⁹ J. Convenient for atomic and particle physics.
Visible photons have 1.8–3.1 eV; a hydrogen ionisation is 13.6 eV.
It's an energy unit, not a voltage.
Potential energy of charges
Energy stored in a system of charges.
U = kq₁q₂/rThe energy to assemble point charges is U = kq₁q₂/r for a pair, and the sum over all pairs for many. Positive for like charges (repulsion).
Three equal charges at triangle corners: 3 pair terms.
Count each pair only once.
Capacitor
Device storing charge and energy.
C = Q/VTwo conductors separated by an insulator store charge ±Q; the capacitance C = Q/V is the charge stored per volt, measured in farads.
A 100 μF capacitor at 12 V holds 1.2 mC.
Typical capacitors are μF to pF.
Parallel-plate capacitor
Capacitance depends on area and gap.
C = ε₀A/dC = ε₀A/d for plate area A and gap d. Increasing area or decreasing gap increases capacitance.
A = 0.01 m², d = 1 mm: 88.5 pF.
Valid when d is small compared with plate size.
Capacitors in series
Reciprocals add.
1/C = 1/C₁+1/C₂Series capacitors store the same charge and their voltages add: 1/C = Σ1/Cᵢ, so the total is smaller than any.
Two 4 μF: 2 μF.
Opposite of resistors.
Capacitors in parallel
Capacitances add.
C = C₁+C₂Parallel capacitors share the same voltage and charges add: C = ΣCᵢ.
4 μF + 6 μF = 10 μF.
Bigger area equivalently.
Energy in a capacitor
Energy stored in the field.
U = ½CV²The stored energy U = ½CV² = Q²/2C = ½QV. It sits in the electric field between the plates.
100 μF at 12 V: 7.2 mJ.
Not QV, which is twice the energy.
Energy density of field
Energy per volume.
u = ½ε₀E²The energy per unit volume of an electric field is u = ½ε₀E². Fields in space carry energy, even in a vacuum.
E = 10⁶ V/m: 4.4 J/m³.
Applies to any electric field.
Dielectrics
Insulators that increase capacitance.
C = κC₀An insulator between plates increases capacitance by factor κ (dielectric constant) and reduces the field by κ. It also raises the breakdown voltage.
Paper κ ≈ 3.7; water 80.
At constant charge voltage drops; at constant voltage charge increases.
Polarisation
Alignment of dipoles inside a dielectric.
In an electric field the charges in a dielectric shift slightly or its molecular dipoles align, producing an internal field opposing the applied one. The net field is reduced to E₀/κ.
Polar molecules rotate; non-polar ones are stretched.
Polarisation is proportional to the field for linear materials.
Dielectric breakdown
Insulation fails above a critical field.
At a critical field (air 3 MV/m) electrons are torn from atoms and a spark forms. Materials with high breakdown strength are used in capacitors.
Lightning occurs over kilometres.
Breakdown fields vary with humidity and shape.
Spherical capacitor
Capacitance of concentric spheres.
C = 4πε₀ab/(b−a)Concentric spheres of radius a and b have C = 4πε₀ab/(b − a). With b → ∞ the isolated sphere has C = 4πε₀a.
Earth's capacitance ≈ 710 μF.
The inner sphere carries the charge.
Cylindrical capacitor
Capacitance per unit length of coaxial cable.
C/L = 2πε₀/ln(b/a)Coaxial cylinders have C/L = 2πε₀/ln(b/a), the model for coaxial cables.
a = 1 mm, b = 3 mm: about 51 pF/m.
Edge effects ignored.
Van de Graaff generator
Builds up high voltages by charge transfer on a belt.
A moving belt carries charge to a hollow metal dome, where it spreads to the outside surface, raising the potential to millions of volts. Used for accelerating particles and demonstrations.
Makes hair stand up.
Charge builds until air breaks down.
Lightning and discharge
Charge separation in clouds produces huge potentials.
Cloud charge separation creates potentials of 10⁸ V; air breakdown forms a leader channel and a return stroke carrying 30 kA. Lightning rods provide a safe path.
Typical strike delivers 1–5 GJ.
Thunder is the shock wave of heated air.
Photocopiers and electrostatic precipitators
Practical uses of electrostatic forces.
A photocopier charges a drum, light discharges blank areas, toner sticks to the charged image, and is transferred to paper. Precipitators charge smoke particles and collect them on plates.
Removing fly ash from power station exhaust.
Both use Coulomb forces on small particles.
Millikan oil drop experiment
Measured the charge of one electron.
Charged oil drops suspended between plates (qE = mg) showed all charges are multiples of e, giving e = 1.6×10⁻¹⁹ C.
Radius found by terminal velocity with Stokes' law.
Won the 1923 Nobel prize.
Motion of charge in uniform field
A charged particle accelerates along field lines.
a = qE/mA charge q of mass m accelerates uniformly a = qE/m. An electron in a 1000 V/m field accelerates at 1.8×10¹⁴ m/s².
Path in a perpendicular field is a parabola.
Ignore gravity for electrons.
Cathode ray tube
Electron beams steered by electric fields.
Electrons accelerated through a potential and deflected by plate fields strike a phosphor screen. Old TVs and oscilloscopes used it.
Deflection the deflection is proportional to the plate voltage.
The beam must travel in a vacuum.
Method of images
Solve problems with conductors by adding fictitious charges.
A charge near a grounded conducting plane produces the same field as if there were an opposite image charge behind the plane. It makes boundary problems solvable.
A +q at distance d from a plane is attracted with F = kq²/(2d)².
Valid only outside the conductor.
Poisson's and Laplace's equations
Differential equations for potential.
∇²V = −ρ/ε₀∇²V = −ρ/ε₀ (Poisson) relates potential to charge density; with ρ = 0 it is Laplace's equation. Solutions are unique given boundary conditions.
Used in numerical field calculations.
Potential has no local maxima in empty space.
Multipole expansion
Expand potential into monopole, dipole and quadrupole terms.
At large distance the potential of any charge distribution expands into monopole (1/r), dipole (1/r²), quadrupole (1/r³) terms. The first nonzero term dominates.
Neutral molecules show dipole fields.
Each term falls faster than the one before.
Boundary conditions at interfaces
Fields change in defined ways across surfaces.
Across a charged surface the normal component of E jumps by σ/ε₀, while the tangential component is continuous. These conditions fix the fields in problems with dielectrics and conductors.
Surface of a conductor: E_tangential = 0.
D (not E) normal component has jump σ_free.
Electrostatic energy of a sphere
Energy of a uniformly charged sphere.
U = 3Q²/20πε₀RA uniformly charged sphere has U = 3Q²/(20πε₀R) = (3/5)kQ²/R. A classical electron radius arises by setting this equal to mc².
r_e = 2.8×10⁻¹⁵ m.
Includes self-energy, not just pair energy.
Earnshaw's theorem
Charges cannot be held in stable equilibrium by electrostatic forces alone.
No collection of static point charges can be held in stable equilibrium by electrostatic forces alone (Laplace's equation has no minima). Stability needs other forces, such as motion or quantum effects.
Atoms are stable thanks to quantum mechanics.
Fails with dynamic fields (ion traps).
Electric field of a ring
On axis the field varies with distance.
E = kQx/(x²+R²)^{3/2}On the axis of a ring of charge Q and radius R, E = kQx/(x² + R²)^{3/2}; zero at the centre and ≈ kQ/x² far away.
Q = 1 nC, R = x = 5 cm: E = 9×10⁹·10⁻⁹·0.05/(0.005)^{3/2} ≈ 1.3 kV/m.
Perpendicular components cancel by symmetry.
Electric field of a disc
Integrate rings to get the disc's field.
Integrate rings: E = (σ/2ε₀)(1 − x/√(x² + R²)). As R → ∞ it approaches the infinite sheet result σ/2ε₀.
Close to the centre the disc acts like a plane.
Check the limits.
Frequently asked questions
Why does a balloon stick to a wall after rubbing on hair?
Rubbing transfers electrons, leaving the balloon with a net charge. It induces opposite charges in the wall's surface, and the attraction holds the balloon in place.
What is the difference between electric field and electric potential?
Electric field is force per unit positive charge and is a vector. Electric potential is energy per unit positive charge and is a scalar, measured in volts.
What does a capacitor store?
A capacitor stores electric energy in the field between two conductors, with equal and opposite charges. It does not store charge in the sense of a battery, and the stored energy is one half of C times V squared.
Why does a balloon stick to a wall after rubbing?
Rubbing charges the balloon, and the charge induces an opposite charge on the wall surface, so the two attract.
Is the electric field always strong near a charge?
The field is strongest close to a charge and falls off with the square of distance, so it becomes weak quickly as you move away.
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