IB DP Physics · Theme D · Fields

Electric and magnetic fields.

D.2 · Charge, Coulomb’s law, electric fields, magnetic field patterns and electric potential
Standard level and higher level

SL + HL · 8 hours Additional HL · 6 hours

The journey

Four parts.

Part 01 · SL + HL

Electric charge

Two types, Coulomb’s law, charging, Millikan

Part 02 · SL + HL

Electric fields

Field strength, field lines, parallel plates

Part 03 · SL + HL

Magnetic fields

Magnets, wires, coils and solenoids

Part 04 · HL only

Electric potential

Potential energy, potential and equipotentials

Slides with a terracotta edge and an HL only flag are additional higher level. Every other slide is for SL and HL.

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Part 01 · SL + HL

Electric charge.

Two types of charge, how bodies become charged, and the force between them

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Positive and negative

Two types of charge.

Like charges repel. Unlike charges attract
Charge is measured in coulombs, C
Protons are positive, electrons negative, with the same size of charge:
\(e = 1.60 \times 10^{-19} \,\mathrm{C}\)
A neutral body has equal amounts of both

FORCES BETWEEN CHARGES

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Why only two types

Every electrical experiment ever done can be explained with two kinds of charge that cancel when they are combined. Which one is called positive is a historical accident: Benjamin Franklin chose the names long before the electron was found, and the electron ended up with the negative sign.

Neutral does not mean empty

A copper coin contains about \(10^{24}\) electrons and exactly as many protons, so its total charge is zero. Charging an object means moving a tiny fraction of its electrons on or off. One coulomb is a huge charge by everyday standards: a rubbed balloon carries less than a millionth of a coulomb.

In solids only electrons move. A body becomes positive by losing electrons, never by gaining protons.

A conservation law

Charge is conserved.

The total charge of an isolated system never changes
Charge can be moved from one body to another, but not created or destroyed

Rubbing

\(0 = (-q) + (+q)\)

Rod and cloth start neutral and end with equal and opposite charges

Touching

\((+6) + (-2) = +4 \,\mathrm{nC}\)

Two identical spheres touch and share the total equally: +2 nC each

Circuits

Junctions

The current into a junction equals the current out of it

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

The spheres, step by step

The total charge is \(+6 + (-2) = +4 \,\mathrm{nC}\). When identical conducting spheres touch, they end with equal charges, so each takes \(+2 \,\mathrm{nC}\). The total is still \(+4 \,\mathrm{nC}\). If the spheres were different sizes the charge would not split equally, but the total would still be +4 nC.

It holds even when particles are created

In beta-minus decay a neutron (charge 0) becomes a proton (+e), an electron (−e) and an antineutrino (0). New particles appear, but the total charge is zero before and after. No process has ever been found that changes the total charge.

Add charges with their signs. A common slip is to add the sizes: +6 nC and −2 nC do not make 8 nC.

Moving electrons

Charging by friction and by contact.

RUBBING A ROD WITH A CLOTHPRESS → TO RUB

Friction: rubbing moves electrons from one insulator to the other
The two end with equal and opposite charges
Contact: touch a charged body to a conductor and charge is shared
An insulator holds its charge in place. In a conductor it spreads

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Which way the electrons go

It depends on the pair of materials. Polythene holds electrons more strongly than a woolen cloth, so it takes electrons and becomes negative. Acetate or glass rubbed with the same cloth gives up electrons and becomes positive. Rubbing only brings the surfaces into close contact over a large area; it is not the rubbing itself that creates the charge.

Why you cannot charge a metal rod held in your hand

Metals and your body are conductors. Any charge that the rod gains flows through you to the ground at once. Hold the metal with an insulating handle and it charges just as well as plastic.

Writing that “positive charges move onto the rod”. The positive charges are protons, fixed in the nuclei. Only electrons transfer.

Charging without touching

Charging by induction.

A METAL SPHERE ON AN INSULATING STANDPRESS → FOR EACH STAGE

The sphere ends with the opposite charge to the rod, and the rod loses none of its own. Earthing (grounding) gives electrons a path to or from the ground.

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

What happens at each stage

The negative rod repels the free electrons in the metal to the far side: the sphere is still neutral overall but its charge is separated. Connecting it to earth lets those repelled electrons escape further, into the ground. Breaking the earth connection first traps the shortage of electrons on the sphere. Only then is the rod taken away, and the positive charge spreads evenly over the surface.

Why the order matters

If the rod is removed before the earth connection is broken, the electrons simply flow back up from the ground and the sphere ends neutral.

What earth is

The Earth is so large that it can supply or accept electrons without its own potential changing. Touching the sphere with a finger works as an earth connection.

With a positive rod everything reverses: electrons flow up from the earth onto the sphere, and it ends negative.

The story · Paris, 1785

Background story · not assessed

Coulomb’s torsion balance.

Painted portrait of Charles-Augustin de Coulomb in a dark coat with a powdered wig, holding a brass instrument — **Charles-Augustin de Coulomb**, military engineer

Engraved plate from 1785 showing a tall glass cylinder with a thin vertical tube on top. Inside, a light horizontal rod with a small ball at one end hangs from a fine thread, near a second fixed ball — His **torsion balance**, from the 1785 memoir

A light rod hung from a fine thread, with a charged ball at one end
A second charged ball pushed it round. The twist of the thread measured the force
Halving the distance needed four times the twist
Published in 1785. Cavendish used the same idea for gravity in 1798

An inverse square law, like Newton’s for gravity.

Portrait: Louis Hierle, 1894, public domain. Balance: Charles-Augustin de Coulomb, 1785, public domain.

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

The force between two point charges

Coulomb’s law.

\(F = k \dfrac{q_{1} q_{2}}{r^{2}}\)

Along the line joining the charges
\(k = 8.99 \times 10^{9} \,\mathrm{N} \,\mathrm{m}^{2} \,\mathrm{C}^{-2}\) in a vacuum
An inverse square law, for point charges

TWO POINT CHARGESEQUAL AND OPPOSITE FORCES

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Reading the sign

Put the charges in with their signs. A positive answer (two like charges) means repulsion; a negative answer (unlike charges) means attraction. Many people prefer to calculate the size of the force from the sizes of the charges and then decide the direction by thinking: both methods are accepted.

How strong it is

Compare it with gravity for the electron and proton in a hydrogen atom. The electric attraction is \(8.2 \times 10^{-8} \,\mathrm{N}\). The gravitational attraction between the same two particles is \(3.6 \times 10^{-47} \,\mathrm{N}\), smaller by a factor of about \(10^{39}\). Gravity wins on the scale of planets only because matter is almost exactly neutral, so the electric forces cancel.

Charges are usually given in μC or nC, and distances in cm. Convert to C and m before you start.

A charged sphere acts as a point charge at its center, for anything outside it, as a uniform sphere does for gravity.

Worked example

The force between two small charged spheres.

\(q_{1} = +3.0 \,\mathrm{\mu C}\) · \(q_{2} = -5.0 \,\mathrm{\mu C}\) Centers \(r = 0.20 \,\mathrm{m}\) apart, in air \(F = k \dfrac{q_{1} q_{2}}{r^{2}}\) \(F = \dfrac{8.99 \times 10^{9} \times 3.0 \times 10^{-6} \times 5.0 \times 10^{-6}}{0.20^{2}}\) \(F = \class{res}{3.4 \,\mathrm{N}}\), attractive

Sizes of the charges in the equation; direction from the signs
\(\mathrm{\mu C} = 10^{-6} \,\mathrm{C}\) and \(\mathrm{nC} = 10^{-9} \,\mathrm{C}\)
Each sphere feels 3.4 N, towards the other

Air is so close to a vacuum that the same \(k\) is used.

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Charges inside a material

The Coulomb constant and permittivity.

\(k = \dfrac{1}{4 \pi \varepsilon_{0}}\)

\(\varepsilon_{0} = 8.85 \times 10^{-12} \,\mathrm{C}^{2} \,\mathrm{N}^{-1} \,\mathrm{m}^{-2}\): the permittivity of free space
In a material, replace \(\varepsilon_{0}\) by its permittivity \(\varepsilon\)
Larger \(\varepsilon\): smaller force

The same two spheres, now in water \(\varepsilon = 80 \varepsilon_{0}\) \(F = \dfrac{1}{4 \pi \varepsilon} \dfrac{q_{1} q_{2}}{r^{2}}\) \(F = \dfrac{F_\mathrm{air}}{80} = \dfrac{3.4}{80} = \class{res}{0.042 \,\mathrm{N}}\)

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Why a material weakens the force

The molecules of an insulating material are distorted or turned by the field of the charges, so that their own positive and negative ends line up against it. Their fields partly cancel the original one. Water molecules do this particularly well, which is why water has such a high permittivity.

Why this matters in chemistry

In a crystal of salt the sodium and chloride ions are held together by electric attraction. In water that attraction is 80 times weaker, and thermal motion is enough to pull the ions apart. That is why salt dissolves in water but not in oil.

Check that \(\frac{1}{4 \pi \times 8.85 \times 10^{-12}}\) gives \(8.99 \times 10^{9}\) on your calculator. Both constants are in the data booklet.

Evidence that charge is quantized

Millikan’s oil drop experiment.

ONE CHARGED OIL DROPPRESS → TO SWITCH ON THE FIELD

Adjust the voltage until the drop hangs at rest: \(qE = mg\)
Measure \(q\) for many drops
Every charge is a whole-number multiple of one value, \(e\)

Charge is quantized: \(q = ne\).

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

What “quantized” means

Charge comes in lumps. A drop can carry 1, 2, 3 or more extra electrons, but never 2.5. Millikan’s drops gave charges such as \(3.2\), \(4.8\), \(8.0\) and \(1.6 \times 10^{-19} \,\mathrm{C}\). No drop ever had a charge between these steps, and the size of one step is the charge of a single electron.

How the mass was found

A drop is far too small to weigh. With the field switched off it falls at a steady terminal velocity, where its weight is balanced by viscous drag and buoyancy. Measuring that velocity through a microscope gives the radius of the drop, and so its mass. The drops picked up their charge from friction in the spray nozzle, or from air ionized by X-rays.

To find \(e\) from data: look for the largest value that divides exactly into every measured charge.

E.2 tells the story of Millikan’s other famous experiment, on the photoelectric effect.

Worked example

How many extra electrons on the drop?.

Mass \(4.9 \times 10^{-15} \,\mathrm{kg}\), at rest between plates \(V = 500 \,\mathrm{V}\) · plates \(d = 5.0 \,\mathrm{mm}\) apart \(qE = mg\) with \(E = \dfrac{V}{d}\) \(q = \dfrac{mgd}{V} = \dfrac{4.9 \times 10^{-15} \times 9.8 \times 5.0 \times 10^{-3}}{500}\) \(q = \class{res}{4.8 \times 10^{-19} \,\mathrm{C}}\)

Number of extra electrons \(n = \dfrac{q}{e}\) \(n = \dfrac{4.8 \times 10^{-19}}{1.60 \times 10^{-19}} = \class{res}{3}\)

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

The story · Chicago, 1909–1913

Background story · not assessed

Millikan, Fletcher and the oil drops.

Museum display of a dull metal drum about the size of a bucket, with small windows and taps in its side, and a round brass plate with a ring of bolts held above it — **Oil drop apparatus** of about 1916, with one of its brass plates lifted out

A drop that evaporates changes its mass while it is being measured
In 1909 Robert Millikan and his student Harvey Fletcher used an oil that hardly evaporates
The 1913 paper gave \(e = 1.592 \times 10^{-19} \,\mathrm{C}\), within 0.6% of today’s value
Millikan alone received the Nobel Prize, in 1923

The first measurement of the charge of a single electron.

Apparatus: Daderot, Museum of Science and Industry, Chicago, CC0.

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Quick check

Electric charge.

01A rod becomes positive when rubbed. What has it gained or lost?

It has lost electrons

02Identical spheres with +8 nC and −2 nC touch. Charge on each afterwards?

\(\frac{8 - 2}{2} = +3 \,\mathrm{nC}\)

03The distance between two charges is tripled. What happens to the force?

It falls to one ninth

04A sphere is charged by induction with a negative rod. Its final charge?

Positive

05What did Millikan’s experiment show about charge?

It is quantized: multiples of \(e\)

06Could a drop carry \(4.0 \times 10^{-19} \,\mathrm{C}\)?

No: that is 2.5\(e\)

PRESS → TO REVEAL EACH ANSWER

Part 02 · SL + HL

Electric fields.

Field strength, field lines and the uniform field between parallel plates

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Force per unit charge

Electric field strength.

\(E = \dfrac{F}{q}\)

The force per unit charge on a small positive test charge at that point
A vector, in N C⁻¹
Force on any charge placed there: \(F = qE\)

Direction

The force on a positive charge

A negative charge is pushed the opposite way to the field

Round a point charge

\(E = k \dfrac{Q}{r^{2}}\)

From Coulomb’s law with \(q_{2}\) as the test charge

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

The same idea as in D.1

Gravitational field strength is force per unit mass; electric field strength is force per unit charge. Both describe the place, not the object you put there. The one new feature is the sign: mass is always positive, but charge is not, so the force on a charge can be along the field or against it.

A worked case

At a point 0.30 m from a charge of +4.0 nC, \(E = \frac{8.99 \times 10^{9} \times 4.0 \times 10^{-9}}{0.30^{2}} = 400 \,\mathrm{N} \,\mathrm{C}^{-1}\), pointing away from the charge. An electron placed there feels \(1.60 \times 10^{-19} \times 400 = 6.4 \times 10^{-17} \,\mathrm{N}\), towards the charge.

Using the test charge in \(E = k \frac{Q}{r^{2}}\). \(Q\) is the charge that makes the field. The test charge canceled.

Drawing a field

Electric field lines.

Show the direction of the force on a positive charge
Start on positive charge, end on negative charge
Never cross
Closer together: a stronger field

RADIAL FIELDS OF POINT CHARGES

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Why density means strength

Draw 12 lines from a point charge. At any distance, all 12 lines pass through a sphere centered on the charge. The area of that sphere grows as \(r^{2}\), so the number of lines through each square meter falls as \(\frac{1}{r^{2}}\), exactly as the field strength does. The picture of lines spreading out is the inverse square law in visual form.

What field lines are not

They are not the path a charge would follow. A charge released from rest starts off along the line, but once it is moving its momentum carries it across curved lines. A field line shows the direction of the force at each point, and nothing more.

Sketching: equal angles between lines from a point charge, arrows on every line, and the number of lines in proportion to the size of the charge.

Fields add as vectors

The field of two point charges.

UNLIKE CHARGESLINES JOIN THEM

LIKE CHARGESZERO FIELD MIDWAY

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

How the patterns arise

At every point, the field of each charge is worked out as if the other were not there, and the two are added as vectors. Close to either charge its own field is much the stronger, so the lines there look radial. Further away the two fields are of similar size and the lines bend.

The midpoint

Between two equal like charges the fields are equal and opposite at the center, so the field there is zero: no line passes through that point. Between unlike charges the two fields at the center point the same way, from the positive charge to the negative one, and they add. For +4.0 nC and −4.0 nC placed 0.60 m apart, each gives 400 N C⁻¹ at the midpoint, so the field there is 800 N C⁻¹.

Lines must leave and meet each charge evenly spaced and at right angles to its surface. Far from two unlike charges of equal size, the field becomes very weak.

Inside and outside

The field of a charged conducting sphere.

The charge spreads over the surface
Inside: no field at all
Outside: the field of a point charge at the center
Lines meet the surface at right angles

A POSITIVE SPHERE

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Why the charge goes to the surface

The extra charges repel one another and are free to move through the metal, so they get as far apart as they can: onto the outside surface, evenly spread on a sphere.

Why there is no field inside

If there were a field inside the metal, the free electrons would feel a force and move. They would keep moving until their new arrangement canceled the field. Within a tiny fraction of a second the charges settle, and when nothing is moving the field inside must be zero. This is true for a solid sphere and for a hollow one.

Why the lines are at right angles

Any component of the field along the surface would push charge along the surface. At rest, there can be none.

A hollow metal box shields its inside from outside electric fields in the same way. A car body protects its passengers in a lightning strike.

Parallel plates

The uniform field between parallel plates.

\(E = \dfrac{V}{d}\)

\(V\): potential difference between the plates. \(d\): their separation
Same strength and direction everywhere between the plates
\(\mathrm{V} \,\mathrm{m}^{-1}\) is the same unit as \(\mathrm{N} \,\mathrm{C}^{-1}\)

OPPOSITELY CHARGED PLATESEDGE EFFECTS

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Where the equation comes from

Move a charge \(q\) from one plate to the other. The force on it is \(qE\) all the way, so the work done is \(qEd\). By the definition of potential difference the work is also \(qV\). Setting them equal gives \(E = \frac{V}{d}\).

Why the field is uniform

Each plate carries an even sheet of charge. Close to a sheet that is large compared with your distance from it, its field does not weaken as you move away: the charge is further off, but more of it pulls in nearly the same direction. Between the plates the fields of the two sheets point the same way and add. The force on a charge is the same at every point between the plates, so a charge there has a constant acceleration, like a mass in the uniform gravitational field near the ground.

The edges

Near the edges the lines bulge outwards and are further apart: the field is weaker and no longer uniform. Calculations assume plates that are large compared with their separation, so that the edges can be ignored.

Draw the lines between the plates with a ruler, evenly spaced, starting and ending exactly on the plates, with one curved line at each edge.

Worked example

An electron between charged plates.

\(V = 600 \,\mathrm{V}\) · plates \(d = 3.0 \,\mathrm{cm}\) apart \(E = \dfrac{V}{d} = \dfrac{600}{0.030} = \class{res}{2.0 \times 10^{4} \,\mathrm{V} \,\mathrm{m}^{-1}}\) \(F = qE = 1.60 \times 10^{-19} \times 2.0 \times 10^{4}\) \(F = \class{res}{3.2 \times 10^{-15} \,\mathrm{N}}\)

Electron mass \(9.11 \times 10^{-31} \,\mathrm{kg}\) \(a = \dfrac{F}{m} = \dfrac{3.2 \times 10^{-15}}{9.11 \times 10^{-31}}\) \(a = \class{res}{3.5 \times 10^{15} \,\mathrm{m} \,\mathrm{s}^{-2}}\) Towards the positive plate

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Why the weight is ignored

The weight of an electron is \(9.11 \times 10^{-31} \times 9.8 = 8.9 \times 10^{-30} \,\mathrm{N}\). The electric force here is larger by a factor of \(10^{14}\). For electrons, protons and ions in electric fields, gravity can always be left out. For an oil drop, which has billions of times more mass, it cannot.

The force is the same everywhere

Because the field is uniform, the answer does not depend on where the electron is between the plates. Its acceleration is constant, so the equations of motion from A.1 apply. D.3 uses this to follow a charged particle through the plates.

Sending the electron along the field. Field lines show the force on a positive charge: the electron accelerates the opposite way.

Two inverse square laws

Gravitational and electric fields compared.

	Gravitational	Electric
Acts on	Mass	Charge
Force law	\(F = G \dfrac{m_{1} m_{2}}{r^{2}}\)	\(F = k \dfrac{q_{1} q_{2}}{r^{2}}\)
Field strength	\(g = \dfrac{F}{m}\), in N kg⁻¹	\(E = \dfrac{F}{q}\), in N C⁻¹
Direction	Always attractive	Attractive or repulsive
Uniform field	Near a planet’s surface	Between parallel plates

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

What is the same

Both forces act at a distance, both obey an inverse square law for point bodies, and both are described by a field strength that is a force per unit of the property the force acts on. A uniform sphere behaves as a point at its center in both cases. Learn one and you have most of the other.

What is different

There is only one kind of mass, so gravity always attracts and can never be shielded. There are two kinds of charge, so electric forces can attract or repel, and equal amounts of both kinds cancel. The constants differ enormously: \(G\) is about \(10^{-10}\) and \(k\) about \(10^{10}\) in SI units.

Using the same words for both (field strength, field line, and at HL potential and equipotential) is deliberate: the mathematics is identical.

Quick check

Electric fields.

01Define electric field strength.

Force per unit charge on a small positive test charge

02A 2.0 μC charge feels a force of 0.010 N. Field strength?

\(\frac{0.010}{2.0 \times 10^{-6}} = 5.0 \times 10^{3} \,\mathrm{N} \,\mathrm{C}^{-1}\)

03Which way do field lines point near a negative charge?

Towards it

04Plates 2.0 cm apart have 100 V across them. Field strength?

\(\frac{100}{0.020} = 5.0 \times 10^{3} \,\mathrm{V} \,\mathrm{m}^{-1}\)

05What is the field inside a charged conducting sphere?

Zero

06Where between parallel plates is the field not uniform?

At the edges

PRESS → TO REVEAL EACH ANSWER

Part 03 · SL + HL

Magnetic fields.

The field patterns of a bar magnet, a straight wire, a circular coil and a solenoid

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

A bar magnet

Magnetic field lines.

Show the direction of the force on a north pole: the way a compass points
Outside a magnet: from N to S
Closer together: a stronger field
Always closed loops: they continue inside the magnet

THE FIELD OF A BAR MAGNET

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

How this differs from an electric field

Electric field lines begin on positive charges and end on negative ones. Nobody has ever found a separate north or south pole: cut a magnet in half and you get two smaller magnets, each with both poles. So magnetic field lines have nowhere to begin or end. They run round in closed loops, out of the north pole, round to the south pole and back through the magnet.

The Earth

A compass needle is a small magnet, and its north pole points towards geographic north. Since unlike poles attract, the magnetic pole near the Earth’s geographic north is a magnetic south pole.

Magnetic field strength is also called magnetic flux density. Its symbol is \(B\) and its unit the tesla, T. D.3 defines it from the force on a current.

The story · Copenhagen, 1820

Background story · not assessed

Ørsted’s compass needle.

Painted portrait of Hans Christian Ørsted, a man with gray hair in a dark coat wearing a decoration — **Hans Christian Ørsted**, painted by C. A. Jensen

Engraving of a compass needle on a stand between two horizontal wires held by posts. Arrows show the current in the upper wire and the needle swinging round — A current in the wire **turns the needle** (engraving of 1876)

Electricity and magnetism were thought to be unrelated
In 1820 Ørsted reported that a current in a wire deflects a compass needle nearby
The needle set itself across the wire, not along it: the field circles the wire
Ampère soon showed that two currents exert forces on each other

A moving charge makes a magnetic field.

Portrait: C. A. Jensen, public domain. Engraving: Augustin Privat-Deschanel, 1876, public domain.

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

A current-carrying wire

The field of a straight wire.

Field lines are circles centered on the wire
Further apart further out: the field gets weaker
Right-hand grip rule: thumb along the current, fingers curl the way of the field

Reverse the current and the field reverses.

CURRENT UPWARDS

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Using the grip rule

Imagine holding the wire in your right hand with the thumb pointing the way the conventional current flows, from positive to negative. Your fingers wrap round the wire in the direction of the field lines. For the wire in the diagram, with the current upwards, the field is counterclockwise when you look down from above.

Why the circles spread out

The field strength round a long straight wire is proportional to \(\frac{1}{r}\): twice as far from the wire, half the field. It is also proportional to the current. This is not an inverse square law, because the source is a long line, not a point.

Using the left hand, or the direction of electron flow. The rule is for the right hand and conventional current.

Looking along the wire

Dots and crosses.

Dot: out of the page, towards you
Cross: into the page, away from you
Out of the page: field counterclockwise
Into the page: field clockwise

A WIRE SEEN END ON

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Shaping the field

A circular coil and a solenoid.

FLAT CIRCULAR COILCROSS-SECTION

AIR-CORE SOLENOIDCROSS-SECTION

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

The coil

Each short piece of the loop makes circles of field round itself. Inside the loop all these circles point the same way, through the coil, so they add up to a strong field there. Outside the loop they partly cancel.

The solenoid

A solenoid is many loops side by side. Their fields add inside to give a field that is strong, parallel to the axis and nearly uniform, which is why solenoids are used wherever a uniform magnetic field is needed. Outside, the pattern is the same as that of a bar magnet. The field is stronger for a larger current and for more turns on each meter of length.

Which end is north?

Grip the solenoid in your right hand with the fingers curling the way the current flows round the turns. Your thumb points to the north pole, the end the field lines leave from.

Looking at one end of a solenoid: current counterclockwise means that end is a north pole; clockwise means south.

Only these four patterns are required: bar magnet, straight wire, circular coil and air-core solenoid.

Quick check

Magnetic fields.

01Which way do field lines run outside a bar magnet?

From the north pole to the south pole

02What shape are the field lines round a long straight wire?

Circles centered on the wire

03A current flows into the page. Direction of the field?

Clockwise

04Describe the field inside a solenoid.

Strong and uniform, along the axis

05How do magnetic field lines differ from electric ones?

They are closed loops, with no start or end

06The current in a wire is reversed. What happens to its field?

It reverses direction

PRESS → TO REVEAL EACH ANSWER

Part 04 · Additional higher level

Electric potential.

Potential energy and potential · the potential gradient
Work done on a charge · equipotential surfaces

D.2 ELECTRIC AND MAGNETIC FIELDS · ADDITIONAL HL

Energy of two charges

Electric potential energy.

\(E_\mathrm{p} = k \dfrac{q_{1} q_{2}}{r}\)

The work done to assemble the system from infinite separation
Zero at infinite separation
Keep the signs of the charges

Like charges

\(E_\mathrm{p}\) is positive

Work must be done to push them together: they would fly apart

Unlike charges

\(E_\mathrm{p}\) is negative

They attract: work must be done to pull them apart

D.2 ELECTRIC AND MAGNETIC FIELDS · ADDITIONAL HL

Compare with gravity

Gravitational potential energy, \(-G \frac{m_{1} m_{2}}{r}\), is always negative because gravity always attracts. Two unlike charges also attract, and putting one positive and one negative charge into \(k \frac{q_{1} q_{2}}{r}\) gives a negative answer in the same way. Two like charges repel, which has no parallel in gravity: their potential energy is positive.

A worked case: the hydrogen atom

For a proton and an electron \(5.3 \times 10^{-11} \,\mathrm{m}\) apart: \(E_\mathrm{p} = \frac{8.99 \times 10^{9} \times (1.60 \times 10^{-19}) \times (-1.60 \times 10^{-19})}{5.3 \times 10^{-11}} = -4.3 \times 10^{-18} \,\mathrm{J}\), which is −27 eV. Adding the electron’s kinetic energy of +13.6 eV gives a total energy of −13.6 eV. The total is negative, so the atom is bound (E.1).

Squaring \(r\), as in Coulomb’s law. Force has \(r^{2}\) underneath; energy has \(r\).

Energy per unit charge

Electric potential.

\(V_\mathrm{e} = \dfrac{kQ}{r}\)

The work done per unit charge in bringing a small positive test charge from infinity to the point
A scalar, in volts. Zero at infinity
Potentials of several charges add as numbers

POTENTIAL ROUND A POINT CHARGE

D.2 ELECTRIC AND MAGNETIC FIELDS · ADDITIONAL HL

Reading the two curves

To bring a positive test charge towards a positive charge you must push against a repulsion, so you do positive work: the potential is positive and rises as you get closer. Towards a negative charge the test charge is pulled in and the field does the work: the potential is negative.

The volt

One volt is one joule per coulomb. The potential difference across a cell in a circuit is the same quantity as here: the difference in the electric potential between two points. A point at 180 V is one where each coulomb of positive charge would have 180 J of potential energy.

Dropping the sign of \(Q\). Potential is a scalar but it does have a sign, and it matters when you add the potentials of several charges.

Worked example

The potential near two charges.

P is 0.10 m from \(Q_{1} = +2.0 \,\mathrm{nC}\) and 0.20 m from \(Q_{2} = -3.0 \,\mathrm{nC}\) \(V_\mathrm{e} = \dfrac{k Q_{1}}{r_{1}} + \dfrac{k Q_{2}}{r_{2}}\) \(V_{1} = \dfrac{8.99 \times 10^{9} \times 2.0 \times 10^{-9}}{0.10} = +180 \,\mathrm{V}\) \(V_{2} = \dfrac{8.99 \times 10^{9} \times (-3.0 \times 10^{-9})}{0.20} = -135 \,\mathrm{V}\) \(V_\mathrm{e} = 180 - 135 = \class{res}{+45 \,\mathrm{V}}\)

No directions, no components: just add
It does not matter where P is, only how far it is from each charge

Field strengths at P would have to be added as vectors.

D.2 ELECTRIC AND MAGNETIC FIELDS · ADDITIONAL HL

Joules and electronvolts

Work done in moving a charge.

\(W = q \Delta V_\mathrm{e}\)

Depends only on the potentials at the start and end, not on the path
Electronvolt: the work done when a charge \(e\) moves through 1 V
\(1 \,\mathrm{eV} = 1.60 \times 10^{-19} \,\mathrm{J}\)

An electron, from rest, moves through +500 V \(\Delta E_\mathrm{k} = -q\Delta V_\mathrm{e} = \class{res}{+500 \,\mathrm{eV}}\) \(= 500 \times 1.60 \times 10^{-19} = 8.0 \times 10^{-17} \,\mathrm{J}\) \(v = \sqrt{\dfrac{2 E_\mathrm{k}}{m}} = \sqrt{\dfrac{2 \times 8.0 \times 10^{-17}}{9.11 \times 10^{-31}}}\) \(v = \class{res}{1.3 \times 10^{7} \,\mathrm{m} \,\mathrm{s}^{-1}}\)

D.2 ELECTRIC AND MAGNETIC FIELDS · ADDITIONAL HL

Who does the work?

\(W = q \Delta V_\mathrm{e}\) is the work an outside agent must do to move the charge without changing its speed. For the electron, \(q\) is negative and \(\Delta V_\mathrm{e}\) is positive, so \(W\) is negative: no one has to push. For the freely accelerating electron, \(q = -e\) and \(\Delta E_\mathrm{p} = q\Delta V_\mathrm{e} = -500 \,\mathrm{eV}\). Conservation gives \(\Delta E_\mathrm{k} + \Delta E_\mathrm{p} = 0\), hence \(\Delta E_\mathrm{k} = -q\Delta V_\mathrm{e} = +500 \,\mathrm{eV}\). Always take \(\Delta V_\mathrm{e} = V_\mathrm{f} - V_\mathrm{i}\).

Why the electronvolt is convenient

An electron accelerated through 500 V gains 500 eV. A particle with charge \(2e\), such as an alpha particle, gains 1000 eV through the same voltage. The number of electronvolts is the number of volts times the charge in units of \(e\), with no powers of ten to carry.

Putting an energy in eV straight into \(\frac{1}{2} m v^{2}\). Convert to joules first.

Linking the two descriptions

Field strength is the potential gradient.

\(E = -\dfrac{\Delta V_\mathrm{e}}{\Delta r}\)

The field points from high potential to low potential
Potential changing quickly with distance: a strong field
Between plates: \(E = \dfrac{400}{0.040} = 1.0 \times 10^{4} \,\mathrm{V} \,\mathrm{m}^{-1}\)

PLATES 4.0 cm APARTEQUIPOTENTIALS EVERY 1.0 cm

D.2 ELECTRIC AND MAGNETIC FIELDS · ADDITIONAL HL

The parallel plates are the simplest case

The potential falls steadily from 400 V at the positive plate to 0 V at the negative plate, by 100 V for every centimeter. The gradient is the same everywhere, so the field is uniform, and \(E = \frac{V}{d}\) from the SL part is this equation for a constant gradient.

Why the minus sign

The potential decreases in the direction of the field, so \(\Delta V_\mathrm{e}\) is negative when \(\Delta r\) is taken along the field. The minus sign makes \(E\) come out positive in that direction. A positive charge is pushed “downhill” in potential, as a ball rolls downhill in height.

On a graph of potential against distance, the field strength is minus the gradient. On a graph of field strength against distance, the area under the curve is the fall in potential along the field.

Mapping a field with potential

Equipotential surfaces.

Join points at the same potential
No work is done moving a charge along one
Always at right angles to the field lines
Round a point charge: spheres. Between plates: planes

A POSITIVE POINT CHARGESTEPS OF 100 V

D.2 ELECTRIC AND MAGNETIC FIELDS · ADDITIONAL HL

Why the circles spread out

The potential is \(\frac{kQ}{r}\), so halving the potential means doubling the distance. The surfaces at 300 V and 200 V are close together; those at 200 V and 100 V are three times as far apart. Closely spaced equipotentials mean a large potential gradient, and so a strong field: near the charge.

Why no work is done

\(W = q \Delta V_\mathrm{e}\), and along an equipotential \(\Delta V_\mathrm{e} = 0\). In terms of force: the field, and so the force on the charge, is at right angles to the surface, and a force at right angles to the motion does no work.

To turn a map of equipotentials into field lines, draw curves that cross every equipotential at right angles, with arrows pointing from higher potential to lower.

Recognizing the patterns

Equipotentials of two and four charges.

UNLIKE CHARGESWITH FIELD LINES

FOUR CHARGESEQUIPOTENTIALS ONLY

D.2 ELECTRIC AND MAGNETIC FIELDS · ADDITIONAL HL

How to read the two-charge map

Close to each charge the equipotentials are nearly circles, positive round the positive charge and negative round the negative one. Exactly halfway between, every point is the same distance from both charges, so the two potentials cancel: the mid-plane is the 0 V equipotential. Every field line crosses every dashed curve at a right angle.

The four charges

The same reasoning gives two planes of zero potential, between each positive charge and its negative neighbors. You are expected to recognize patterns like these and to sketch field lines on them, not to calculate them.

Equipotentials of different values can never cross: a point cannot have two different potentials. The two zero planes of the four charges do meet, at the center, but they share the same value, and the field there is zero.

Solid or hollow

Field and potential of a charged sphere.

Outside: as a point charge at the center
Inside: the field is zero, so the potential does not change
The whole sphere is at the surface potential \(\dfrac{kQ}{R}\)

\(Q = 2.0 \,\mathrm{nC}\) · \(R = 0.10 \,\mathrm{m}\) \(V_\mathrm{e} = \dfrac{kQ}{R} = \class{res}{180 \,\mathrm{V}}\) throughout

AGAINST DISTANCE FROM THE CENTER

D.2 ELECTRIC AND MAGNETIC FIELDS · ADDITIONAL HL

Why the potential is constant inside

\(E = -\frac{\Delta V_\mathrm{e}}{\Delta r}\). Inside the conductor \(E = 0\), so the potential gradient is zero and the potential cannot change from one point to another. No work is needed to move a charge around inside. The whole conductor, surface and interior, is one equipotential.

Solid and hollow spheres behave alike

In both, the charge is on the outer surface and the field inside is zero, so the graphs are the same. Outside, the equipotentials are spheres centered on the conductor, as for a point charge.

The field at the surface

\(E = \frac{kQ}{R^{2}} = \frac{8.99 \times 10^{9} \times 2.0 \times 10^{-9}}{0.10^{2}} = 1800 \,\mathrm{V} \,\mathrm{m}^{-1}\). This is the gradient of the lower graph just outside the surface.

Drawing the potential falling to zero inside because the field does. Zero field means zero change in potential.

Quick check · additional higher level

Electric potential.

01Is electric potential a scalar or a vector?

A scalar

02Sign of the potential energy of a proton and an electron?

Negative: they attract

03A proton moves through a potential difference of 2000 V. Energy gained?

2000 eV = \(3.2 \times 10^{-16} \,\mathrm{J}\)

04The potential falls by 50 V over 2.0 mm. Field strength?

\(\frac{50}{2.0 \times 10^{-3}} = 2.5 \times 10^{4} \,\mathrm{V} \,\mathrm{m}^{-1}\)

05At what angle do field lines cross equipotentials?

90°

06The surface of a metal sphere is at 300 V. Potential at its center?

300 V

PRESS → TO REVEAL EACH ANSWER

Summary

Electric and magnetic fields on one slide.

Charge

Two types, conserved, quantized

Friction, contact and induction move electrons
Millikan: \(q = ne\)

Coulomb’s law

An inverse square force

\(F = k \frac{q_{1} q_{2}}{r^{2}}\) with \(k = \frac{1}{4 \pi \varepsilon_{0}}\)

Electric field

Force per unit charge

\(E = \frac{F}{q}\). Uniform between plates: \(E = \frac{V}{d}\)
Zero inside a conductor

Magnetic field

Closed loops

Bar magnet, wire, coil, solenoid
Right-hand grip rule for the direction

Potential · HL

Energy per unit charge

\(V_\mathrm{e} = \frac{kQ}{r}\), a scalar
\(W = q \Delta V_\mathrm{e}\), in J or eV

Equipotentials · HL

At right angles to the field

\(E = -\frac{\Delta V_\mathrm{e}}{\Delta r}\)
A conductor is one equipotential

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL

Summary

Key words.

Conservation of chargeTotal charge never changes

QuantizedIn whole multiples of e

InductionCharging without contact

EarthingA path for electrons to the ground

PermittivityHow a medium weakens the force

Electric field strengthForce per unit positive charge

Uniform fieldSame strength and direction everywhere

SolenoidA long coil of many turns

Right-hand grip ruleThumb: current. Fingers: field

Electric potentialWork per unit charge from infinity

ElectronvoltEnergy of charge e through 1 V

Equipotential surfaceJoins points at the same potential

D.2 ELECTRIC AND MAGNETIC FIELDS · SL + HL