PHY132H1/Unit 1

Unit 1 · Introduction to Physics II

Electric Charge & Force

A new fundamental property, and a force law with the same shape as gravity but two crucial differences: it comes in two signs, and it is about $10^{36}$ times stronger. Everything in this course follows from that.

By the end you can

• State the properties of charge, including conservation and quantisation.
• Apply Coulomb's law with correct direction.
• Superpose forces from several charges.
• Explain conductors, insulators and induced charge.

Why it matters & where it comes from

Why it exists

Every force you meet in daily life other than gravity is electromagnetic. The normal force, friction, tension, chemical bonds — all are electric forces between atoms, dressed up.

In the real world

Photocopiers and laser printers move toner with electrostatic attraction. Lightning is charge separation released. Static shocks are the same physics at a smaller scale.

Story & open questions

Coulomb measured the inverse-square law in 1785 with a torsion balance delicate enough to detect forces of a millionth of a newton. Why charge is quantised in units of $e$ — and why the proton's charge exactly balances the electron's — is still unexplained.

Charge

Definition

Charge is a property of matter, measured in coulombs. Its rules:

Two signs. Like charges repel, unlike attract. This is the deep difference from gravity, which only attracts.

Conserved. Charge is never created or destroyed, only moved. Rubbing a balloon transfers electrons; it does not make charge.

Quantised. Every observed charge is a whole multiple of $e = 1.6 \times 10^{-19}$ C.

Rule Conductors and insulators

In a conductor, some electrons move freely. Charge redistributes almost instantly and always ends up on the surface.

In an insulator, charges stay put. Charge remains wherever it was placed.

Example Why a charged rod attracts neutral paper

The rod's field pulls the paper's electrons slightly toward it and pushes its nuclei slightly away — the paper polarises.

The near side is now oppositely charged and the far side likewise charged. Because force falls off with distance, the near attraction beats the far repulsion.

Net attraction, with no net charge on the paper at all. This is why neutral objects are attracted to charged ones regardless of the charge's sign.

Coulomb's law

Rule The force between two point charges

$F = k\dfrac{|q_1 q_2|}{r^2}$ with $k = 8.99 \times 10^9$ N·m$^2$/C$^2$

The force acts along the line joining them: repulsive for like signs, attractive for unlike. Use the formula for the magnitude and settle direction by inspection.

Do not put signs into the formula

Substituting negative charges and hoping the sign comes out right leads to confusion in two dimensions. Compute the magnitude from absolute values, then draw the arrow. This is the single most useful habit in the unit.

Example Comparing with gravity

Two protons a distance $r$ apart.

Electric repulsion: $ke^2/r^2$. Gravitational attraction: $Gm_p^2/r^2$.

The ratio is about $10^{36}$, and $r$ cancels entirely — both are inverse-square, so the comparison is the same at every distance.

Gravity dominates at large scales only because bulk matter is almost exactly neutral. Electric forces cancel; gravity cannot, because there is no negative mass.

Example A numerical case

Charges of $+3\,\mu$C and $-2\,\mu$C, $0.05$ m apart.

$F = (8.99 \times 10^9)(3 \times 10^{-6})(2 \times 10^{-6}) / (0.05)^2$

$= (8.99 \times 10^9)(6 \times 10^{-12}) / (2.5 \times 10^{-3}) = 21.6$ N.

Attractive, since the signs differ. Note how large that is for microcoulomb charges — a reminder that a coulomb is an enormous amount of charge.

Superposition

Rule Forces add as vectors

With several charges present, the force on one is the vector sum of the forces from each other charge taken separately. Each pair is computed as though the others were absent.

Example Three charges in a line

Charges of $+2\,\mu$C at $x = 0$, $+3\,\mu$C at $x = 0.1$ m, and $-1\,\mu$C at $x = 0.2$ m. Find the force on the middle one.

From the left charge: repulsion, so pointing $+x$. Magnitude $k(2)(3) \times 10^{-12}/(0.1)^2 = 5.39$ N.

From the right charge: attraction, so also pointing $+x$. Magnitude $k(3)(1) \times 10^{-12}/(0.1)^2 = 2.70$ N.

Both point the same way, so the total is $8.09$ N in the $+x$ direction. Deciding each direction by inspection made the sum trivial.

Example Two dimensions

When the charges are not collinear, resolve each force into components, add the components, then recombine — exactly the PHY131H1 Unit 1 procedure.

Never add magnitudes. Two $5$ N forces at right angles give $7.1$ N, not $10$ N.

Common pitfalls

• Putting signs into Coulomb's law instead of deciding direction by inspection.
• Adding force magnitudes rather than components.
• Forgetting to square the distance, or using diameter for radius.
• Leaving charges in microcoulombs. Convert to coulombs first.
• Thinking a neutral object cannot be attracted. Polarisation makes it so.

Practice

Work each fully before opening the answer. Original exercises written for this guide in the style of the course's problem sets.

Practice 1 · Direction

Two negative charges. Attract or repel?

Answer

Repel. Like signs always repel, whichever sign they are.

Practice 2 · Distance

Halving the separation does what to the force?

Answer

Quadruples it. The force goes as $1/r^2$.

Practice 3 · Magnitude

Find the force between $+1\,\mu$C and $+1\,\mu$C at $0.1$ m.

Answer

$F = (8.99 \times 10^9)(10^{-12})/(0.01) = 0.899$ N, repulsive.

Practice 4 · Neutral attraction

Why does a charged comb pick up paper?

Answer

The paper polarises. Its near side becomes oppositely charged and, being closer, is attracted more strongly than the far side is repelled.

Practice 5 · Equilibrium point

Charges $+q$ at $x = 0$ and $+4q$ at $x = 1$. Where is the field-free point on the line between them?

Solution

Set the magnitudes equal: $q/x^2 = 4q/(1-x)^2$, so $(1-x) = 2x$ and $x = 1/3$. Closer to the smaller charge, as it must be.

Practice 6 · Counting electrons

How many electrons make $-1\,\mu$C?

Answer

$10^{-6}/(1.6 \times 10^{-19}) = 6.25 \times 10^{12}$ — over six trillion, for a charge that is small by laboratory standards.

Practice 7 · Conductor

Where does excess charge sit on an isolated metal sphere?

Answer

Entirely on the outer surface. Mutual repulsion drives the charges as far apart as possible, and Unit 2 proves the interior field is zero.

Practice 8 · Two dimensions

Charges $+q$ at the two ends of a base and $+q$ at the apex of an equilateral triangle. Which way is the force on the apex charge?

Solution

Directly away from the midpoint of the base. The horizontal components from the two base charges cancel by symmetry and the vertical ones add, so the net force is $2F\cos 30$ upward where $F$ is either individual force.

Expanded topic checklist

Course coverage. This unit is part of the complete PHY132H1 topic map. It should cover the following items before you move on:

• Charge. Two signs, conservation, quantisation.
• Coulomb's law. Magnitude, direction by inspection, and the comparison with gravity.
• Superposition. Vector addition of forces from several charges.
• Materials. Conductors, insulators, polarisation and induced charge.

Textbook/source alignment

The calendar's topic list opens with "electricity". Check the syllabus on Quercus for the assigned reading and problems; this material is normally the first two weeks.

Core idea map

Charge produces force; force falls off as $1/r^2$; forces from many charges add as vectors. Unit 2 recasts all of this in terms of a field, which removes the need to keep asking "force on what?".

Study routine. Draw the force arrows before computing anything. Direction settled by inspection and magnitude by formula is faster and far less error-prone than trying to do both at once.

Deep practice extensions

Use these after the main practice set. They push on definitions and edge cases rather than routine calculation.

Extension 1 · Two signs

What does having two signs of charge allow that gravity cannot?

Guide

Cancellation. Bulk matter is neutral because positive and negative balance, which is why gravity dominates at large scales despite being far weaker.

Extension 2 · Quantisation

What is the smallest observed free charge?

Guide

$e = 1.6 \times 10^{-19}$ C. Quarks carry thirds of it but are never found alone.

Extension 3 · Direction

Why not substitute signed charges into Coulomb's law?

Guide

Because the sign of the result does not encode a direction in two dimensions. Compute the magnitude and draw the arrow.

Extension 4 · Polarisation

Can a neutral object experience a net electric force?

Guide

Yes, if it polarises in a non-uniform field. The nearer induced charge is attracted more than the farther one is repelled.

Back to IndexAll chapters Unit 02 →The electric field