Unit 2 · Introduction to Physics II

The Electric Field

A change of viewpoint rather than of physics. Instead of asking what force one charge exerts on another, ask what condition the first charge creates in space. The field is real, it carries energy, and in Unit 7 it detaches and travels as light.

By the end you can

Define the field and compute it from point charges.
Draw and read field lines.
Use Gauss's law on symmetric charge distributions.
Explain why the field inside a conductor is zero.

Why it matters & where it comes from

Why it exists

Coulomb's law describes action at a distance, which troubled everyone who used it. The field replaces it with something local: the charge alters space around it, and another charge responds to the space where it actually is.

In the real world

Faraday cages work because the field inside a conductor is zero — which is why you are safe in a car during a lightning strike, and why an MRI room is lined with metal.

Story & open questions

Faraday introduced field lines in the 1830s as a picture, being largely self-taught and uncomfortable with algebra. Maxwell then turned the picture into equations, and the field became the more fundamental object — a rare case of an intuition outlasting the formalism it replaced.

Definition

The electric field at a point is the force per unit charge a small positive test charge would feel there:

$E = \dfrac{F}{q}$, measured in N/C.

The field exists whether or not anything is there to feel it. Placing a charge $q$ in a field gives $F = qE$.

Rule Field of a point charge

$E = k\dfrac{|Q|}{r^2}$, pointing away from a positive charge and toward a negative one.

Fields from several charges superpose as vectors, exactly as forces did.

Direction depends on the test charge's sign

The field direction is fixed by the source. But the force on a charge placed in it depends on that charge's sign: positive charges are pushed along the field, negative charges against it. Confusing the two is the commonest error in the unit.

Example Field on the axis of a dipole

Charges $+q$ and $-q$ separated by $d$. At a point on the perpendicular bisector:

Each contributes a field of the same magnitude $kq/r^2$.

The components along the bisector cancel; the components parallel to the dipole axis add.

The net field points from positive to negative, parallel to the axis — the opposite of what many expect from a sketch.

Field lines

Rule How to read them

Lines start on positive charges and end on negative ones.

The field direction at a point is the tangent to the line, not the line itself.

Density of lines indicates field strength.

Lines never cross — the field has one direction at each point.

Lines meet a conductor's surface perpendicularly.

Example Why perpendicular at a surface

Suppose a line met the surface at an angle. It would then have a component along the surface.

That component would push the conductor's free charges sideways, and they would move.

They keep moving until no such component remains. So in equilibrium the field must be perpendicular — the argument is entirely about what the charges would do.

Gauss's law

Definition

Flux measures how much field passes through a surface. For a uniform field through a flat area, $\Phi = EA\cos\theta$, with $\theta$ between the field and the surface's normal.

Rule Gauss's law

The total flux through any closed surface depends only on the charge enclosed:

$\Phi = \dfrac{Q_{\text{enclosed}}}{\varepsilon_0}$ with $\varepsilon_0 = 8.85 \times 10^{-12}$ C$^2$/(N·m$^2$)

Charges outside contribute nothing: their field enters the surface and leaves again, and the two cancel exactly.

Example Field of a charged sphere

A sphere carrying total charge $Q$, at a distance $r$ outside it.

Choose a spherical surface of radius $r$ centred on the charge. By symmetry the field has the same magnitude everywhere on it and points radially.

Flux is then $E \times 4\pi r^2$, and Gauss's law gives $E = \dfrac{Q}{4\pi\varepsilon_0 r^2} = \dfrac{kQ}{r^2}$.

Identical to a point charge at the centre. A uniformly charged sphere acts, from outside, exactly like all its charge concentrated at its middle — the same result as for gravity, for the same geometrical reason.

Example Inside a conductor

Take a Gaussian surface entirely within the metal of a charged conductor in equilibrium.

If the field were nonzero anywhere inside, free charges would move — contradicting equilibrium. So $E = 0$ throughout the metal.

Zero field means zero flux, so by Gauss's law the enclosed charge is zero. Any excess charge must therefore sit on the surface.

This is the Faraday cage. A hollow conductor shields its interior from any external field, which is why sensitive electronics live in metal boxes.

Gauss's law is always true, rarely useful

It holds for every closed surface and every charge distribution. But it only lets you solve for $E$ when symmetry makes the field constant over the surface — spherical, cylindrical or planar. Without symmetry it is true and useless.

Common pitfalls

• Confusing field direction with force direction on a negative charge.
• Thinking field lines are paths a charge follows. They are not trajectories.
• Including charges outside the Gaussian surface. Only enclosed charge counts.
• Applying Gauss's law where there is no symmetry.
• Forgetting that $E = 0$ inside conductor material but not inside a hollow cavity containing charge.

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 · Field from a charge

Find the field $0.2$ m from a $+5\,\mu$C charge.

Answer

$E = (8.99 \times 10^9)(5 \times 10^{-6})/0.04 = 1.12 \times 10^6$ N/C, pointing away.

Practice 2 · Force on a negative charge

An electron sits in a field pointing east. Which way is the force?

Answer

West. Negative charges feel a force opposite the field.

Practice 3 · Crossing lines

Why can field lines never cross?

Answer

A crossing point would have two field directions at once, which is meaningless.

Practice 4 · Enclosed charge

A closed surface encloses $+3\,\mu$C and $-1\,\mu$C, with $+5\,\mu$C outside. What is the flux?

Answer

$\Phi = (2 \times 10^{-6})/\varepsilon_0 = 2.26 \times 10^5$ N·m$^2$/C. The outside charge contributes nothing.

Practice 5 · Inside a shell

What is the field inside a uniformly charged spherical shell?

Solution

Zero. A Gaussian sphere inside encloses no charge, and symmetry forces the field to be uniform on it — so it must vanish. The shell's own charge contributes nothing anywhere inside.

Practice 6 · Infinite sheet

Use Gauss's law to find the field of an infinite plane with surface charge density $\sigma$.

Solution

Take a cylinder through the sheet with end caps of area $A$. Flux is $2EA$ — two caps — and the enclosed charge is $\sigma A$. So $E = \sigma/(2\varepsilon_0)$, independent of distance. A remarkable result: the field does not weaken as you move away.

Practice 7 · Faraday cage

Why is a car a safe place in a lightning storm?

Answer

The metal body is a conductor, so the field inside it is zero and the charge flows around the outside to the ground. The rubber tyres are irrelevant.

Practice 8 · When Gauss fails

Why can Gauss's law not give the field of an arbitrary lump of charge?

Answer

It is still true, but without symmetry the field varies over the surface and cannot be pulled out of the flux integral. You learn the total flux and nothing about the field at any particular point.

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:

The field. Force per unit charge, and the distinction between field direction and force direction.
Field lines. The five rules and what each encodes.
Gauss's law. Flux, enclosed charge, and the three symmetric cases where it solves for $E$.
Conductors. Zero interior field, surface charge, and shielding.

Textbook/source alignment

The calendar's "electricity" topic covers this. Check the syllabus on Quercus for assigned problems; Gauss's law questions are usually restricted to the three symmetric geometries.

Core idea map

The field is what a charge does to space. Lines picture it, Gauss's law computes it when symmetry allows, and conductors rearrange until their interior field vanishes. Unit 3 adds the energy view of the same thing.

Study routine. Before any Gauss's law problem, ask what symmetry the charge has. If the answer is none, the law will not help and another method is needed.

Deep practice extensions

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

Extension 1 · Field versus force

Does the field direction depend on the test charge?

Guide

No. The field is fixed by the sources. Only the force on a placed charge depends on that charge's sign.

Extension 2 · Flux

What determines the flux through a closed surface?

Guide

Only the charge enclosed. External charges contribute equal inward and outward flux, which cancels.

Extension 3 · Conductors

Why is the field zero inside conductor material?

Guide

Any field would move the free charges, and they move until it is cancelled. Equilibrium requires zero.

Extension 4 · Usefulness

When does Gauss's law actually solve a problem?

Guide

When spherical, cylindrical or planar symmetry makes the field constant over a chosen surface, so it factors out of the flux.

← Unit 01Electric charge & force Unit 03 →Electric potential