Difficulty: Introductory | Prerequisites: Basic algebra, vector addition, familiarity with electric charge (positive/negative conventions)
Big picture: This is the opening chapter of electricity and magnetism. Before you can do anything with circuits, fields, or flux, you need a solid grip on how charges behave when they are stationary: how objects become charged, how charges redistribute on conductors, and how to calculate the force between point charges. Everything that follows in the course (Gauss's law, potential, capacitance) builds directly on the ideas here. If you missed the first few lectures, start here.
Objects can be charged by friction, contact, or induction. Charging by induction uses a nearby charged object and a temporary ground connection to leave a net charge on a conductor without ever touching it. Coulomb's law gives the force between point charges, and because force is a vector, you must add contributions from multiple charges component by component.
Electric charge
A fundamental property of matter, measured in coulombs (C). Comes in two varieties: positive and negative. Like charges repel; opposite charges attract. In simple terms, charge is what makes objects push or pull on each other electrically.
Conductor
A material (typically a metal) in which charges are free to move. In simple terms, electrons can slide around inside a conductor until they reach equilibrium.
Electroscope
A device used to detect the presence and sign of electric charge. It has a metal knob connected to thin metal leaves; when charge spreads to the leaves, they repel each other and diverge.
Charging by induction
A process that gives an object a net charge without direct contact with the charging body. A charged rod is brought near a conductor, the conductor is grounded (allowing charge to flow in or out), the ground is removed, then the rod is removed. The conductor is left with a net charge opposite in sign to the rod.
Grounding
Connecting a conductor to an effectively infinite reservoir of charge (the Earth). Think of it as opening a tap: charges flow in or out until the conductor reaches the same potential as the ground.
Coulomb's law
The magnitude of the electrostatic force between two point charges is F = k|q₁q₂|/r², where k = 8.99 × 10⁹ N·m²/C². The force acts along the line joining the charges, attractive for opposite signs, repulsive for like signs. In simple terms, it is the electric version of Newton's law of gravitation, but it can push as well as pull.
Superposition principle
The net force (or field) at a point is the vector sum of the individual forces (or fields) from every source charge. Each charge contributes independently; you just add them up, component by component.
Linear charge density (λ)
Charge per unit length, measured in C/m. Used when charge is spread along a wire or curve rather than concentrated at a point.
A positively charged rod is brought near (but does not touch) the metal knob of an electroscope.
Negative charges in the conductor are attracted toward the knob; positive charges are repelled toward the leaves.
The leaves diverge because they both carry the same sign of charge.
While the rod is still nearby, the electroscope is briefly connected to ground.
Negative charges flow from the ground up into the electroscope, attracted by the nearby positive rod.
The leaves partially collapse because excess positive charge has been neutralised.
The ground connection is removed first, then the rod is taken away.
The electroscope now has a net negative charge (it gained electrons from the ground).
With the rod gone, this excess negative charge redistributes evenly, and the leaves diverge again, both negatively charged.
Key point: the final charge on the electroscope is opposite in sign to the rod, even though the rod never touched it.
For two point charges, calculate the magnitude with F = kq₁q₂/r² and determine the direction from the sign of each charge and their geometry.
For three or more charges, find the force on the charge of interest from each source charge individually, then add the x-components together and the y-components together.
When charges are arranged on an x-y plane, the distance r between two charges at coordinates (x₁, y₁) and (x₂, y₂) is r = √[(x₂ − x₁)² + (y₂ − y₁)²].
The x-component of a force along a line at angle θ from the x-axis is F cos θ; the y-component is F sin θ. The angle can also be found from the geometry directly using the displacement components.
Consider Q₁ = +5 μC at (0, 4 cm), Q₂ = +5 μC at (0, −4 cm), and Q₃ = −7 μC at (−3 cm, 0), with a point P at (3 cm, 0).
Electric field at P from Q₁ and Q₂ only:
Q₁ is at distance r = √(3² + 4²) = 5 cm from P. The field from Q₁ points away from Q₁ (positive charge), directed toward the lower-right.
Q₂ is symmetric, at the same distance, with its field pointing toward the upper-right.
The y-components cancel by symmetry. Both x-components point in the positive x-direction.
The net field from Q₁ and Q₂ at P is in the positive x-direction.
Net force on a +1 C test charge at P from all three charges:
Contributions from Q₁ and Q₂ both push the test charge in the +x direction (like charges repel, and the geometry gives net +x after symmetry cancels y).
Q₃ is negative and sits to the left of P, so it attracts the +1 C charge in the −x direction.
The net x-component depends on the magnitudes. With the numbers given, the result is on the order of 10⁷ N, and the sign depends on which contributions dominate.
Coulomb's law:
F = k |q₁ q₂| / r²
where k = 1/(4πε₀) = 8.99 × 10⁹ N·m²/C²
Electric field from a point charge:
E = kq / r²
directed radially away from positive charges, toward negative charges.
Vector components:
Fₓ = F cos θ, F_y = F sin θ
Charging by induction is the principle behind electrostatic precipitators used in coal power plants to remove particulate pollution from exhaust gases. The particles pick up charge by induction and are then attracted to collecting plates.
Coulomb's law, combined with superposition, is the starting point for understanding how every electronic device works, from the charge distributions inside a transistor to the forces between molecules in chemistry.
Students often think charging by induction requires the charged rod to touch the conductor. It does not; the whole point of induction is that charge transfer happens through the ground connection, not through direct contact with the rod.
Students frequently confuse the sign of the final charge. The electroscope ends up with the opposite sign to the rod, because the ground connection allows same-sign charges to leave (or opposite-sign charges to arrive).
When adding forces from multiple charges, students sometimes add magnitudes without accounting for direction. Force is a vector; you must resolve into components and sum each component separately.
Students sometimes forget to convert centimetres to metres when using Coulomb's law. The constant k is in SI units, so distances must be in metres.
⚠️ Charging by induction is a favourite exam question because it tests whether you understand the sequence of steps and the role of grounding. Pay close attention to what happens when the ground is removed before the rod versus after.
⚠️ When the rod is moved away from an electroscope that was charged by induction, the leaves repel because they share the same net charge. The charge is real and permanent (until something else discharges it), not "induced" in the transient sense.
⚠️ For multi-charge Coulomb's law problems, draw a clear diagram, label distances and angles, and resolve into x and y components before touching the calculator. Errors almost always come from geometry, not from the formula itself.
True or false: When a positively charged rod is brought near a grounded electroscope and then the ground is removed before the rod, the electroscope ends up with a net negative charge.
Fill in the blank: Coulomb's law states that the force between two point charges is proportional to the product of the charges and inversely proportional to the ________ of the distance between them.
True or false: If two positive charges are placed on the y-axis, symmetric about the origin, the net electric field they produce at a point on the positive x-axis points in the positive x-direction.
True or false: A conductor in electrostatic equilibrium can have a net electric field inside it.
Fill in the blank: The SI unit of electric charge is the ________.
Q: A positively charged rod is brought near (but does not touch) an electroscope. The electroscope is then briefly grounded and the ground is removed. Finally the rod is taken away. What is the sign of the charge left on the electroscope, and do the leaves diverge after the rod is removed?
A: The electroscope is left with a net negative charge. The leaves diverge after the rod is removed because they share the same negative charge and repel each other.
Q: Is it true that "positive charges will be induced on the electroscope when the rod is moved away" after the induction process described above?
A: No. The electroscope has a net negative charge after the ground is removed. When the rod is taken away, the negative charge simply redistributes evenly. No positive charge is "induced" at that stage.
Q: Three point charges Q₁ = +5 μC at (0, 4 cm), Q₂ = +5 μC at (0, −4 cm) are symmetric about the x-axis. What is the direction of the net electric field they produce at a point on the positive x-axis?
A: Positive x-direction. The y-components from Q₁ and Q₂ cancel by symmetry; the x-components both point away from the charges, which means in the +x direction at a point to the right of the origin.
Q: When computing the force on a test charge from multiple source charges, why can you not simply add the magnitudes of the individual forces?
A: Because force is a vector quantity. The individual forces point in different directions, so you must break each into x and y components, sum the components separately, and then find the resultant magnitude and direction.
This material connects directly to electric fields (the next topic), because E = F/q, so every Coulomb's law problem is also an electric field problem in disguise. It also connects to Gauss's law, which provides a shortcut for finding electric fields in situations with high symmetry, bypassing the vector sums you do here. The idea of superposition introduced here carries through the entire course.
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