Source: Chapter 5
Tags: electric charge, Coulomb's law, electric field, conductors, insulators, induction, electric dipole, superposition, charge conservation, point charge
Difficulty: Introductory
Prerequisites: Basic algebra, trigonometry, vector addition.
This is the opening chapter of the electromagnetism sequence. Everything that follows in PHYS 212, from Gauss's law to circuits to optics, builds on the ideas introduced here: what electric charge is, how charges exert forces on one another, and how those forces are described using the electric field. If you missed this material, every subsequent topic will feel harder than it should. You should already be comfortable with Newton's laws, free-body diagrams, and vector components.
Electric charges come in two types (positive and negative), and the force between them is described by Coulomb's law. Like charges repel, opposite charges attract, and the force drops off with the square of the distance. The electric field is the force-per-unit-charge that a source charge creates in the space around it.
Electric charge
A fundamental property of matter. Comes in two varieties: positive (carried by protons) and negative (carried by electrons). The elementary charge is e = 1.602 x 10⁻¹⁹ C. In simple terms, charge is what makes objects push or pull on each other electrically.
Conservation of charge
In a closed system, the total electric charge never changes. Charges can move around or be redistributed, but the net amount stays the same. Think of it as: you can shuffle charge from one place to another, but you cannot create or destroy it.
Conductor
A material in which electrons move freely. Metals are the classic example. The outermost electrons are loosely bound to any one atom and can migrate through the material. In simple terms, conductors let charge flow.
Insulator
A material in which electrons are tightly bound and cannot move freely. Rubber, glass, and plastic are common examples. In simple terms, insulators block the flow of charge.
Ion
An atom that has gained or lost electrons, giving it a net charge. Ions are never electrically neutral.
Charging by induction
A process in which a charged object is brought near (but does not touch) a conductor, causing a redistribution of charge within the conductor. This creates an electric dipole in the conductor without any direct contact. Think of it as: the charged object reaches across the gap and rearranges the conductor's charges at a distance.
Electric dipole
A pair of equal and opposite charges separated by a small distance. The dipole moment is p = qd, where q is the magnitude of each charge and d is the separation.
Coulomb's law
The law governing the electric force between two point charges. F = k(q₁q₂)/r², where k = 8.99 x 10⁹ N·m²/C². In simple terms, this is the electric version of Newton's law of gravitation: bigger charges and shorter distances mean stronger forces.
Superposition principle
When multiple charges are present, the net force on any one charge is the vector sum of the individual forces from every other charge. You calculate each pair separately and then add the vectors.
Electric field (E)
The force per unit charge at a point in space, created by source charges. E = F/q. Its direction is the direction a positive test charge would be pushed. Think of it as: the electric field is a map of what would happen to a positive charge placed at each point in space.
Electric field lines
Imaginary lines that show the direction and relative strength of the electric field. They point away from positive charges and toward negative charges. The closer together the lines, the stronger the field.
Field line density
The number of field lines per unit area passing through a surface perpendicular to the field. Higher density means a stronger electric field in that region.
Electric forces are either attractive (opposite signs) or repulsive (same signs).
The magnitude of the electric force decreases as the distance between charges increases, following an inverse-square relationship: F proportional to 1/r².
A net charge of zero means the positive and negative charges in an object cancel out exactly.
In an atom, the nucleus holds the protons and neutrons (99% of the mass). Electrons orbit the nucleus.
In a neutral atom, the number of protons equals the number of electrons.
A negative ion has gained electrons. A positive ion has lost electrons.
In conductors, the outermost electrons are loosely bound and can move to neighbouring atoms. This is why metals conduct electricity.
In insulators, electrons are tightly held and cannot flow through the material.
Charging by induction works like this: bring a charged rod near a conducting sphere. The rod's charge repels like charges and attracts unlike charges within the sphere, creating an electric dipole. The sphere is still overall neutral, but charge is redistributed.
Neutral objects can be attracted to any charged object, because the closer induced charge experiences a stronger force than the farther one.
To give the sphere a lasting induced charge: connect it to ground while the rod is nearby (electrons flow in or out through the ground wire), then disconnect from ground, then remove the rod.
For two point charges q₁ and q₂ separated by distance r:
F = k × |q₁ × q₂| / r²
where k = 8.99 x 10⁹ N·m²/C².
If the charges have the same sign, the force is repulsive (along the line connecting them, pushing apart). If opposite, the force is attractive (pulling together).
Newton's third law applies: the force on charge 1 due to charge 2 is equal and opposite to the force on charge 2 due to charge 1.
When more than two charges are present, find the force on the charge of interest from each other charge individually, then add the force vectors (component by component).
Use trigonometry to resolve forces into x and y components, sum them, and find the magnitude and direction of the net force.
The electric field at a point is defined as the force a positive test charge would experience there, divided by the magnitude of that test charge: E = F/q.
Direction: the field points away from positive source charges and toward negative source charges.
Once you know the electric field at a location, the force on any charge q placed there is F = qE.
Mass of an electron: 9.11 x 10⁻³¹ kg. This is relevant when using F = ma to find the acceleration of an electron in a known field.
For extended (continuous) charge distributions, the charge is described by a density:
Linear charge density: λ = Q/L (charge per unit length)
Surface charge density: σ = Q/A (charge per unit area)
Volume charge density: ρ = Q/V (charge per unit volume)
Common configurations and their fields (where ε₀ = 8.85 x 10⁻¹² C²/N·m²):
Infinite line of charge: E = λ / (2πε₀r)
Ring of charge (on axis): E = kQz / (z² + R²)^(3/2)
Disk of charge (on axis): uses integration of rings
Infinite plane of charge: E = σ / (2ε₀), uniform, independent of distance
Two infinite parallel planes (equal and opposite charge): fields add between the plates (E = σ/ε₀) and cancel outside
Field lines originate on positive charges and terminate on negative charges.
The direction of the field at any point is tangent to the field line through that point.
Field line density (lines per unit area) represents the magnitude of the field.
Field lines never cross.
Dipole moment: p = qd (magnitude of each charge times the separation vector).
Torque on a dipole in a uniform field: τ = pE sin θ, where θ is the angle between p and E.
Induced dipoles occur when an external field distorts the charge distribution of an atom or molecule, even if the atom has no permanent dipole.
Quantity | Formula |
|---|---|
Coulomb's law | F = k |q₁q₂| / r², k = 8.99 x 10⁹ N·m²/C² |
Electric field (point charge) | E = kQ / r² |
Force from field | F = qE |
Dipole moment | p = qd |
Torque on dipole | τ = pE sin θ |
Infinite line field | E = λ / (2πε₀r) |
Infinite plane field | E = σ / (2ε₀) |
Two parallel planes | E = σ / ε₀ (between), E = 0 (outside) |
Charging by induction is the principle behind electrostatic precipitators, which remove soot and particulates from industrial exhaust. The electric field concept is foundational to how touchscreens detect your finger: the screen senses changes in the local electric field when a conductor (you) approaches.
Students often think that a neutral object cannot be attracted to a charged one. It can, through induction, because the near side of the neutral object acquires an induced charge closer to the charged object.
Students frequently forget that Coulomb's law gives the magnitude of the force and that the direction must be determined separately by looking at the signs of the charges.
The electric field is not the same as the electric force. The field exists in space regardless of whether a test charge is there. The force depends on the specific charge placed in the field.
When using superposition with multiple charges, students sometimes add force magnitudes instead of adding vector components. Always resolve into x and y first.
⚠️ Coulomb's law problems with three or more charges almost always require vector decomposition. Practice resolving forces into components.
⚠️ Know the distinction between conductors and insulators, and be able to describe the induction charging process step by step.
⚠️ Be able to sketch electric field lines for simple configurations: point charges, dipoles, parallel plates.
⚠️ The relationship E = F/q is definitional. If the exam gives you a field, you must be able to find the force on any charge placed in it, and vice versa.
True or false: A positively charged rod brought near a neutral conductor will repel all electrons in the conductor to the far side.
False. It attracts electrons to the near side.
Fill in the blank: The SI unit of electric charge is the ________.
Coulomb (C).
True or false: The electric field between two parallel plates with equal and opposite charge is zero.
False. The field between the plates adds up to σ/ε₀. The field is zero outside.
Fill in the blank: The electric force between two charges is proportional to 1/________.
r² (the square of the distance).
True or false: Electric field lines can cross each other.
False. If they crossed, the field would have two directions at one point, which is impossible.
Q: A hydrogen atom has a proton (+e) and an electron (-e) separated by 5.29 x 10⁻¹¹ m. Calculate the electric force between them.
A: F = k|q₁q₂|/r² = (8.99 x 10⁹)(1.602 x 10⁻¹⁹)² / (5.29 x 10⁻¹¹)² ≈ 8.2 x 10⁻⁸ N. The force is attractive.
Q: Two charges, q₁ = +3 μC and q₂ = -5 μC, are separated by 0.2 m. What is the magnitude of the force between them?
A: F = (8.99 x 10⁹)(3 x 10⁻⁶)(5 x 10⁻⁶) / (0.2)² = 3.37 N. The force is attractive because the charges have opposite signs.
Q: An electric field of 500 N/C points to the right. What force does a -2 μC charge experience in this field?
A: F = qE = (2 x 10⁻⁶)(500) = 1 x 10⁻³ N. Because the charge is negative, the force is to the left (opposite the field direction).
Q: Explain why a neutral object can be attracted to a charged object.
A: The charged object induces a separation of charge in the neutral object through induction. The side of the neutral object nearest the charged object acquires an opposite induced charge, which is closer and therefore experiences a stronger force. The net force is attractive.
This chapter connects directly to Gauss's law (Ch. 6), which provides a more powerful method for finding electric fields in symmetric situations. The electric field concept introduced here also leads into electric potential (Ch. 7), where we assign an energy-per-charge value to each point in space. Every circuit and electromagnetic phenomenon in the rest of the course depends on the force law and field concept established here.
electric charge, Coulomb's law, Coulomb force, electrostatic force, electric field, field lines, charge conservation, conductor, insulator, induction, polarisation, electric dipole, dipole moment, superposition, point charge, inverse square law, elementary charge, e = 1.602e-19, permittivity of free space, ε₀, charge density, linear charge density, surface charge density, volume charge density, PHYS 212