Physics Fundamentals

Electrostatics: Electric Charge, Coulomb’s Law, Fields and Capacitance

Electrostatics is the study of electric charges that are stationary or nearly stationary. It explains why rubbed materials attract small pieces of paper, why sparks can jump from a doorknob, how lightning forms, and how devices such as photocopiers, air cleaners and capacitors work.

What Is Electrostatics?

All matter is made of atoms. Atoms contain positively charged protons, negatively charged electrons and uncharged neutrons. An object becomes electrically charged when it gains or loses electrons.

An object with equal amounts of positive and negative charge is electrically neutral. If it has more electrons than protons, it has a negative charge. If it has fewer electrons than protons, it has a positive charge.

Definition: Electrostatics

Electrostatics is the branch of physics concerned with electric charges at rest and the forces, fields and energy associated with them.

Electric Charge

Electric charge is a physical property that causes matter to experience electric forces. Charge is represented by the symbol q or Q.

Positive Charge

A positive charge occurs when an object has lost electrons. A proton has a positive elementary charge.

Negative Charge

A negative charge occurs when an object has gained electrons. An electron has a negative elementary charge.

Basic Rules of Electric Charge

Like charges repel one another.
Unlike charges attract one another.
Charge is measured in coulombs, symbol C.
Charge can be transferred from one object to another.
Charge is conserved: it cannot be created or destroyed in an isolated system.

Quantization of Charge

Q = ne

Here, Q is the total charge in coulombs, n is the number of elementary charges, and e is the magnitude of the charge of one proton or electron.

Q: charge, measured in coulombs (C)
n: a whole number with no unit
e: elementary charge, approximately 1.60 × 10⁻¹⁹ C

This relationship is used when finding the number of electrons transferred or checking whether a charge is a whole-number multiple of the elementary charge.

Example: Number of Electrons in a Charge

Given: An object has a charge of −3.20 × 10⁻¹⁹ C.

Formula: n = |Q|/e

Substitution:

n = (3.20 × 10⁻¹⁹ C)/(1.60 × 10⁻¹⁹ C)

n = 2

Final answer: The object has gained 2 electrons. The negative sign shows that the charge is negative.

How Objects Become Charged

In ordinary charging processes, electrons move between objects. Protons remain bound inside atomic nuclei and do not normally move from one material to another.

Charging by Friction

When two different materials are rubbed together, electrons may transfer from one material to the other. One object becomes negatively charged and the other becomes positively charged.

Charging by Contact

When a charged object touches a neutral conductor, electrons can move between them. The neutral object may acquire the same type of charge as the original object.

Charging by Induction

A charged object brought near a conductor can cause charges inside the conductor to separate. With suitable grounding, the conductor can be charged without direct contact.

Polarization

Polarization is the separation or rearrangement of positive and negative charge within an otherwise neutral object. A charged comb can attract neutral paper because it polarizes the paper.

Conductors and Insulators

Property	Conductors	Insulators
Movement of charge	Charges can move relatively freely.	Charges are strongly restricted from moving.
Examples	Copper, aluminium and other metals	Plastic, rubber, glass and dry wood
Electrostatic behavior	Excess charge spreads over the surface.	Excess charge tends to remain near where it was placed.
Common use	Electrical wiring and shielding	Wire insulation and protective coverings

In electrostatic equilibrium, the electric field inside an ideal conductor is zero. Any excess charge resides on the outer surface. Charge density is often greater near sharp points, which is why lightning rods use pointed conductors.

Coulomb’s Law

Coulomb’s law gives the electrostatic force between two point charges. The force becomes larger when the charges are larger and becomes smaller as the distance between them increases.

Magnitude of the Electrostatic Force

F = k |q₁q₂|/r²

F: electrostatic force, measured in newtons (N)
q₁ and q₂: the two charges, measured in coulombs (C)
r: distance between the centers of the charges, measured in metres (m)
k: Coulomb constant, approximately 8.99 × 10⁹ N·m²/C² in vacuum

The magnitude formula gives the size of the force. The direction depends on the signs of the charges: the force is repulsive for like charges and attractive for unlike charges.

This formula is used for point charges or objects that can reasonably be treated as point charges, especially when their size is much smaller than the distance between them.

Inverse-Square Relationship

Because force is proportional to 1/r², doubling the distance reduces the force to one-fourth of its original value. Tripling the distance reduces it to one-ninth.

Example: Force Between Two Charges

Given:

q₁ = +2.0 µC = 2.0 × 10⁻⁶ C
q₂ = −3.0 µC = −3.0 × 10⁻⁶ C
r = 0.50 m

Formula: F = k|q₁q₂|/r²

Substitution:

F = (8.99 × 10⁹)(2.0 × 10⁻⁶)(3.0 × 10⁻⁶)/(0.50)²

F = (8.99 × 10⁹)(6.0 × 10⁻¹²)/0.25

F ≈ 0.216 N

Final answer: The electrostatic force is approximately 0.22 N. Because the charges have opposite signs, the force is attractive.

Principle of Superposition

When several charges act on a charge, the total force is the vector sum of the individual forces.

Total Electric Force

F_total = F₁ + F₂ + F₃ + …

Each force must be added with its direction. For forces along one straight line, choose a positive direction and use positive or negative signs. For forces at angles, resolve them into perpendicular components before adding them.

This principle is used whenever more than two charges are present.

Electric Field

An electric field is the region around a charge where another charge experiences an electric force. Electric field is a vector quantity, so it has both magnitude and direction.

Definition: Electric Field

The electric field at a point is the electric force per unit positive test charge placed at that point.

Electric Field from Force

E = F/q

E: electric field strength, measured in newtons per coulomb (N/C)
F: force on the test charge, measured in newtons (N)
q: test charge, measured in coulombs (C)

This formula is used when the force on a known test charge is given.

Electric Field of a Point Charge

E = k|Q|/r²

E: electric field strength in N/C
Q: source charge in C
r: distance from the source charge in m
k: Coulomb constant in N·m²/C²

The field points away from a positive source charge and toward a negative source charge. The field of several charges is found by vector addition.

Force on a Charge in an Electric Field

F = qE

Here, F is force in N, q is charge in C and E is electric field strength in N/C. The force is in the direction of the field for a positive charge and opposite the field direction for a negative charge.

Example: Electric Field of a Point Charge

Given: Q = 4.0 µC = 4.0 × 10⁻⁶ C and r = 0.20 m.

Formula: E = k|Q|/r²

Substitution:

E = (8.99 × 10⁹)(4.0 × 10⁻⁶)/(0.20)²

E = (3.596 × 10⁴)/0.040

E ≈ 8.99 × 10⁵ N/C

Final answer: The electric field strength is approximately 9.0 × 10⁵ N/C, directed away from the positive charge.

Electric Field Lines

Electric field lines are diagrams used to represent an electric field.

Field lines point in the direction a positive test charge would move.
Lines point away from positive charges and toward negative charges.
Closer field lines represent a stronger field.
Field lines never cross because the field has only one direction at a particular point.
Field lines meet the surface of a conductor at right angles in electrostatic equilibrium.

Electric Flux and Gauss’s Law

Electric flux describes how much electric field passes through a surface. Gauss’s law is especially useful for calculating electric fields when the charge distribution has high symmetry, such as spherical, cylindrical or planar symmetry.

Electric Flux for a Uniform Field

Φ = EA cos θ

Φ: electric flux, measured in N·m²/C
E: electric field strength in N/C
A: surface area in m²
θ: angle between the electric field and the perpendicular to the surface

Gauss’s Law

Φ_E = Q_enclosed/ε₀

Φ_E: total electric flux through a closed surface
Q_enclosed: net charge inside the surface, in C
ε₀: permittivity of free space, approximately 8.85 × 10⁻¹² C²/(N·m²)

Gauss’s law is used mainly when symmetry makes the electric field easy to determine over a chosen closed surface.

Electric Potential and Potential Difference

Electric potential describes electric potential energy per unit charge. It is a scalar quantity, so it does not have a direction.

Electric Potential

V = U/q

V: electric potential, measured in volts (V)
U: electric potential energy, measured in joules (J)
q: charge, measured in coulombs (C)

One volt is equal to one joule per coulomb: 1 V = 1 J/C.

Potential Due to a Point Charge

V = kQ/r

Here, Q is the source charge in C, r is the distance in m, and k is Coulomb’s constant. This formula gives the potential relative to a reference point at infinity.

Potential Energy of Two Point Charges

U = kQq/r

U is electric potential energy in joules. A positive value indicates that energy must be supplied to bring like charges together from far away. A negative value is associated with an attractive pair of charges under this reference choice.

Electric Field and Potential

Electric field describes force per unit charge, while electric potential describes energy per unit charge. A charge can move from one potential to another and experience a change in electric potential energy.

Capacitors

A capacitor is a device that stores separated electric charge and electrical energy. It usually consists of two conductors separated by an insulating material called a dielectric.

Capacitance

C = Q/V

C: capacitance, measured in farads (F)
Q: magnitude of charge on either plate, measured in C
V: potential difference between the plates, measured in V

Capacitance tells us how much charge a capacitor stores for each volt of potential difference.

Parallel-Plate Capacitor

C = ε₀A/d

For a parallel-plate capacitor with air or vacuum between the plates, A is the plate area in m², d is the separation in m and ε₀ is the permittivity of free space. With a dielectric material, the relationship becomes:

C = κε₀A/d

Here, κ is the dielectric constant, a dimensionless quantity.

Energy Stored in a Capacitor

U = 1/2 CV²

U is stored energy in joules, C is capacitance in farads and V is potential difference in volts. This formula is used to calculate the electrical energy stored in a charged capacitor.

Example: Energy Stored in a Capacitor

Given: C = 20 µF = 20 × 10⁻⁶ F and V = 12 V.

Formula: U = 1/2 CV²

Substitution:

U = 1/2(20 × 10⁻⁶)(12)²

U = 1/2(20 × 10⁻⁶)(144)

U = 1.44 × 10⁻³ J

Final answer: The capacitor stores 1.44 × 10⁻³ J of energy.

Real-Life Applications of Electrostatics

Lightning protection: Lightning rods provide a conducting path that helps protect buildings from electrical discharge.
Photocopiers and laser printers: Electrostatic charges help attract toner particles to selected regions of a page.
Electrostatic precipitators: Charged particles in industrial exhaust can be attracted to oppositely charged collecting plates.
Spray painting: Charged paint droplets are attracted to an oppositely charged object, helping paint reach surfaces more evenly.
Capacitors: Capacitors store electrical energy in electronic circuits, flashes and power systems.
Faraday cages: Conducting enclosures can shield their interiors from external electrostatic fields.

Common Mistakes and Misconceptions

Confusing charge with force: Charge is a property of matter, while force is an interaction between charges.
Forgetting the square on distance: Coulomb’s law contains r², not r.
Ignoring charge signs: The signs determine attraction or repulsion, even when the magnitude calculation uses absolute values.
Using centimetres or microcoulombs without conversion: Convert distance to metres and charge to coulombs before substituting into SI formulas.
Mixing up electric field and force: Electric field is force per unit charge. The force on a particular charge is F = qE.
Thinking a neutral object cannot be attracted: A neutral object can be polarized, allowing attraction by a charged object.
Assuming electrons are created during rubbing: Electrons are transferred from one material to another; total charge remains conserved.
Adding forces as ordinary numbers in every situation: Forces are vectors, so direction must be considered.

Important Points to Remember

There are two types of electric charge: positive and negative.
Like charges repel and unlike charges attract.
Charge is measured in coulombs and is conserved.
Coulomb’s law follows an inverse-square relationship with distance.
Electric field is force per unit positive test charge.
Electric field is a vector; electric potential is a scalar.
Excess charge on a conductor resides on its surface in electrostatic equilibrium.
Capacitors store separated charge and electrical energy.
Always use SI units when applying electrostatic formulas.

Quick Recap

Electrostatics studies stationary electric charges and their effects. Charge may be positive or negative, and it is conserved and quantized in multiples of the elementary charge. Coulomb’s law calculates the force between charges, while the electric field describes the force available per unit charge around a source charge. Electric potential represents energy per unit charge, and capacitors store charge and electrical energy. Understanding these ideas provides the foundation for studying electric circuits, electronics and electromagnetism.