Every electric charge is surrounded by a field that can push or pull on other charges, and the same idea — force per unit something, spreading out from a source — turns up again around every magnet. This workbook builds up electric charge and its behaviour from scratch, introduces Coulomb's law for the force between point charges, and then describes the fields those charges create using field lines and field strength. It finishes with the special case of a uniform field between parallel plates, and the field lines drawn around bar magnets.
By the end of this workbook you should be able to:
state the direction of the force between like and unlike charges, and explain conservation of charge and the quantisation of charge (as shown by Millikan's experiment)
describe how charge is transferred by friction, contact and induction, and explain the role of earthing (grounding)
state and apply Coulomb's law, F = kq₁q₂/r², for point charges
define electric field strength, E = F/q, and sketch electric field lines, relating field line density to field strength
calculate the uniform electric field strength between parallel plates, E = V/d
sketch magnetic field lines around a bar magnet and around the Earth
1. Electric charge
Every atom contains charged particles: positive protons and negative electrons. Normally an object has equal numbers of each, so it has no overall charge — we say it is neutral. If an object gains or loses electrons, it becomes charged.
Key word — charge: a property of matter that causes it to experience a force in an electric field. Charge is measured in coulombs (C).
Two types of charge
There are only two types of electric charge: positive and negative. The rule for the direction of the force between them is simple.
Like charges repel. Unlike charges attract. Two positive charges push each other apart; two negative charges push each other apart; a positive and a negative charge pull towards each other.
Fig. 1.1 Like charges repel; unlike charges attract.
Conservation of charge
Charge cannot be created or destroyed. In any closed system, the total charge stays the same, even when charge moves from one object to another. If one object loses a certain amount of negative charge, another object must gain exactly that amount.
Conservation of charge: the total electric charge of an isolated system is always constant.
Quantisation of charge
Charge does not come in just any amount — it always comes in whole-number multiples of a smallest possible charge, called the elementary charge, e. Every proton carries charge +e and every electron carries charge −e.
e = 1.60 × 10−19 C
This means any charge you measure — however it was produced — will always be a whole-number multiple of 1.60 × 10⁻¹⁹ C. This surprising fact was confirmed experimentally by Robert Millikan in 1909. Workbook 2 in this pair is entirely devoted to that experiment, including a simulation you can run yourself, so we won't go into it further here.
Worked example 1.1
A charged sphere has a charge of −6.4 × 10⁻¹⁹ C. How many extra electrons does it carry?
Answer: number of electrons = charge ÷ e = (6.4 × 10⁻¹⁹) ÷ (1.60 × 10⁻¹⁹) = 4 electrons.
1A metal sphere P, carrying charge +3e, is held near an identical neutral sphere Q. State whether P and Q would attract, repel, or feel no force at this moment, and explain why.
No overall force yet, because Q is neutral — it has no net charge for P's field to act on (although P's field can still induce a slight charge separation inside Q; that is covered in Section 2). Once Q becomes charged, the direction of the force depends on the sign it ends up with.
2Two identical conducting spheres carry charges of +9.0 nC and −3.0 nC. They are touched together and then separated. Use conservation of charge to determine the charge on each sphere afterwards. (Hint: identical conducting spheres share any charge equally.)
Hint: add the two charges together to find the total charge on the pair, then split that total equally between the two identical spheres.
3Explain, in your own words, why the statement "electrons can be created by rubbing two objects together" is incorrect.
Rubbing does not create charge — it only transfers existing electrons from one object's surface to the other. The total charge of the two objects together is unchanged (conservation of charge); one object ends up with extra electrons (negative) and the other with a deficit of electrons (positive).
2. Charging and discharging
There are three ways that charge can be transferred between objects: friction, contact and induction. You need to be able to explain each one in terms of the movement of electrons.
Charging by friction
When two insulators are rubbed together, electrons are transferred from one surface to the other. The material that gains electrons becomes negatively charged; the material that loses electrons becomes positively charged (because it is now left with more protons than electrons).
Fig. 2.1 Friction transfers electrons from one insulator to another.
When friction occurs between two different insulating materials, electrons are transferred from the surface of one to the surface of the other.
Charging by contact
If a charged object physically touches a neutral (or oppositely-charged) conductor, some charge flows between them until it is shared. The object that was charged keeps some of its original charge; the object it touched now carries charge of the same sign.
Charging by induction
Induction charges an object without contact. A charged object is brought close to a neutral conductor. Electrons in the conductor are attracted or repelled, so they redistribute — one side becomes negative, the other positive — even though the conductor's overall charge is still zero. If the conductor is then earthed while the charged object is still nearby, electrons can flow to or from the ground, leaving the conductor with a net charge once the earth connection and the charged object are both removed.
Key word — earthing (grounding): connecting an object to the ground with a conductor so that charge can flow freely between the object and the Earth, leaving the object at 0 V.
4A negatively charged rod is brought close to (but does not touch) a neutral metal sphere on an insulating stand.
a) Use the simulator below to predict how charge redistributes on the sphere. Click on each side of the sphere until it shows the charge you think is correct.
Click the near half and the far half of the sphere to set a charge on each.
Metals contain free electrons that can move within the sphere. Think about which type of charge would be attracted towards a nearby negative rod, and which type would be pushed away from it.
b) The sphere is then earthed briefly while the rod stays in place. Explain what happens.
c) The earth connection is removed, then the rod is taken away. What is the final charge on the sphere?
b) While earthed, the repelled electrons can escape to the ground, since they are being pushed away from the rod and towards the earth wire. c) Once the earth wire is removed (rod still present) the sphere has a deficit of electrons; removing the rod afterwards leaves the sphere with an overall positive charge, spread evenly.
5Aircraft are earthed with a bonding cable before refuelling. Explain, using ideas about friction and charge build-up, why this matters and what could happen if it were skipped.
As fuel and air flow past the aircraft's surfaces, friction transfers charge, so the aircraft can build up a large static charge. Without earthing, a spark could jump between the aircraft and the fuel truck as they reach different potentials, and this spark could ignite fuel vapour. The bonding cable keeps both at the same potential (0 V) so charge can drain away safely instead of sparking.
6A student rubs a balloon on a wool jumper. The balloon becomes negatively charged. State the sign of charge left on the jumper, and explain how you know, without measuring anything.
The jumper must be left positively charged. Friction only transfers electrons — it cannot create charge — so whatever negative charge the balloon has gained must be exactly balanced by a positive charge left behind on the jumper (conservation of charge).
3. Coulomb's law
Coulomb's law tells us how strong the electric force is between two point charges. It was published by Charles-Augustin de Coulomb in 1783.
F = k q1q2 / r²
Coulomb's law: the force, F, between two point charges q₁ and q₂, separated by a distance r, is given by F = kq₁q₂/r², where k is the Coulomb constant, k = 8.99 × 10⁹ N m² C⁻².
Notice the shape of this equation — it is an inverse square law, just like Newton's law of gravitation. If you double the separation, the force becomes four times weaker; if you triple it, the force becomes nine times weaker.
All boards · Light
The Inverse Square Law
Spread the same energy over a sphere and its area grows as r². So whatever you measure at a point — brightness, field strength, force — falls as 1/r². Pick a quantity, then move the detector.
Light intensityI ∝ 1/r²
Where the inverse square law turns up
Any influence that streams outward from a point and isn't absorbed obeys it — across mechanics, fields, waves and nuclear physics, and far beyond the exam spec.
?The law fails when the spreading isn't over a full sphere — a laser beam stays roughly parallel, and a long wire or charged plate spreads over a cylinder or plane, giving 1/r or a constant field instead.
Interactive inverse-square-law simulation — available in the online version of this workbook. See drjonesphysics.com/general-apps.
A positive answer for F means the force is repulsive (both charges have the same sign); a negative answer means the force is attractive (the charges have opposite signs).
Key word — point charge: a charge treated as though it exists at a single point in space. A uniformly charged sphere behaves, from the outside, exactly like a point charge at its centre.
Worked example 3.1
Calculate the force between two point charges of +2.0 × 10⁻⁸ C and −5.0 × 10⁻⁸ C, separated by 4.0 cm in air.
Answer:
F = kq₁q₂/r² = (8.99 × 10⁹) × (2.0 × 10⁻⁸) × (−5.0 × 10⁻⁸) / (0.040)²
F = −5.6 × 10⁻³ N (the negative sign shows the force is attractive)
7Two point charges of +4.0 μC and +6.0 μC are separated by 25 cm in air. Calculate the force between them, and state whether it is attractive or repulsive.
Hint: substitute directly into F = kq₁q₂/r², using k = 8.99×10⁹ N m² C⁻². Convert the separation to metres before squaring it. The sign of both charges tells you whether the force is attractive or repulsive.
8The force between two identical point charges is 8.0 × 10⁻³ N when they are 12 cm apart. Determine the magnitude of each charge.
Hint: since the two charges are identical, Coulomb's law becomes F = kq²/r². Rearrange to make q² the subject, substitute in SI units, then take the square root.
9The force between two point charges separated by 20 cm is 6.0 × 10⁻⁵ N. Predict the new force if the separation is reduced to 5.0 cm, without recalculating the charges.
Hint: you don't need to find the actual charges. Work out the factor by which the separation has changed, then use F ∝ 1/r² to see how the force scales by the square of that factor.
4. Electric fields
A region of space where a charge would experience an electric force is called an electric field. We describe electric fields using two related ideas: field lines (a picture) and field strength (a number).
Electric field strength
E = F / q
Electric field strength, E, is the force per unit charge that a small positive test charge would feel at that point. SI unit: N C⁻¹ (equivalent to V m⁻¹).
Field lines
Field lines are a way of drawing an electric field. Three rules always apply:
Field lines point in the direction of the force on a positive test charge.
Field lines never cross.
The closer together the lines are, the stronger the field — field line density represents field strength.
Fig. 4.1 Radial field lines around an isolated positive charge (left) and negative charge (right). Lines point away from positive charge, towards negative charge.
Worked example 4.1
A charge of +4.0 nC experiences a force of 2.4 × 10⁻⁵ N at a certain point. Calculate the electric field strength at that point.
Answer: E = F/q = (2.4 × 10⁻⁵) / (4.0 × 10⁻⁹) = 6.0 × 10³ N C⁻¹
Drag charges onto the canvas below (or use a preset), then tick Show Field Lines to see the pattern build up. Try recreating the two-equal-positive-charges setup described above and look for the point where the lines cancel.
Interactive electric field lines simulation — available in the online version of this workbook. See drhanburyphysics.com/labs/electric-fields.html.
10A point charge of 5.0 nC feels a field strength of 3.0 × 10⁴ N C⁻¹. Calculate the force on the charge.
Hint: rearrange E = F/q to make F the subject, then substitute the given field strength and charge directly.
11Two diagrams show field lines around a charged object. In diagram A the lines are close together; in diagram B (same type of charge, but drawn further from the object) the lines are much more spread out. Explain what this tells you about how field strength changes with distance from a point charge.
Because field line density represents field strength, lines spreading out further from the charge shows that the field gets weaker with increasing distance — consistent with the inverse-square relationship E = kq/r².
5. The uniform field between parallel plates
A very useful electric field can be made by connecting a potential difference across two parallel metal plates. Between the plates (away from the edges), the field is uniform — the same strength and direction everywhere.
Fig. 5.1 The uniform field between two oppositely charged parallel plates, separated by distance d.
E = V / d
Uniform field strength between parallel plates: E = V/d, where V is the potential difference across the plates and d is the separation between them.
This equation shows that N C⁻¹ and V m⁻¹ are equivalent units for electric field strength.
Worked example 5.1
Two parallel plates are separated by 1.2 cm and connected to a 600 V supply. Calculate the electric field strength between them.
Answer: E = V/d = 600 / 0.012 = 5.0 × 10⁴ V m⁻¹
12Parallel plates 8.0 mm apart produce a field of 2.5 × 10⁵ V m⁻¹. Calculate the potential difference across them.
Hint: rearrange E = V/d to make V the subject, and convert the plate separation to metres before substituting.
13A student halves the plate separation but keeps the same potential difference across a parallel-plate arrangement. Explain what happens to the field strength between the plates, and to the force on a charge placed there.
Since E = V/d and V is unchanged while d halves, E doubles. Because F = Eq, the force on any given charge placed between the plates also doubles.
6. Magnetic field lines
A magnetic field exists anywhere a magnetic force acts. Just like electric fields, we represent magnetic fields on paper using field lines — but the rules for drawing them are slightly different.
Magnetic field lines always form closed loops — they never simply start or stop.
Outside a magnet, field lines point from the north pole to the south pole.
Like electric field lines, they never cross, and are closest together where the field is strongest.
Fig. 6.1 Magnetic field lines around a bar magnet
Permanent magnets and poles
A bar magnet has two poles: a north pole and a south pole. Like poles repel; unlike poles attract — exactly like electric charges. But there is one big difference: an isolated single magnetic pole has never been found. If you cut a bar magnet in half, you don't get a separate N piece and a separate S piece — you get two smaller magnets, each with its own N and S pole.
The Earth's magnetic field
The Earth behaves like a giant, weak bar magnet, which is why a compass needle (a small permanent magnet, free to rotate) lines up with it. The end of a compass needle that points towards geographic North is called its "north pole" — which means it is actually attracted towards a magnetic south pole located near the Earth's geographic North.
14Two bar magnets are placed end to end, north pole facing north pole, a small gap between them. Describe the magnetic field lines you would expect to see in the gap between the two north poles, and state whether there is a point between the poles where the resultant field is zero.
Because like poles repel, the field lines from each north pole curve away from the gap rather than crossing it; there is a point on the line joining the poles, midway between them, where the resultant field is zero.
15Explain why cutting a bar magnet in half does not produce an isolated north pole and an isolated south pole.
Magnetic poles always occur in pairs (dipoles) — a single isolated pole ("monopole") has never been observed. Cutting the magnet simply creates two smaller magnets, each with its own complete N and S pole.
7. Equation summary
Quantity
Equation
Unit
Coulomb's law
F = kq₁q₂/r²
N
Electric field strength
E = F/q
N C⁻¹ (= V m⁻¹)
Field strength, point charge
E = kQ/r²
N C⁻¹
Field strength, parallel plates
E = V/d
V m⁻¹
Coulomb constant
k = 8.99 × 10⁹
N m² C⁻²
Elementary charge
e
1.60 × 10⁻¹⁹ C
8. Mixed practice
These questions draw on several sections at once, the way an exam question might.
16Two identical small conducting spheres, A and B, are mounted on insulating stands. Sphere A carries a charge of +8.0 nC and sphere B is neutral.
a) The spheres are placed 30 cm apart and touched together, then separated again to the same distance. State the charge on each sphere after separation, and name the physics principle you used. b) Calculate the force between the two spheres in their new, separated positions. c) A third, identical neutral sphere C is then brought close to (but not touching) sphere A. Describe, without calculation, how the charge redistributes on sphere C.
Hint: a) conservation of charge means the total +8.0 nC is shared equally between two identical conductors, giving +4.0 nC on each. b) substitute the two new, equal charges and the 30 cm separation into F = kq₁q₂/r². c) this is induction — free electrons in sphere C are attracted towards the nearby positive sphere A, so the near side of C becomes negative and the far side becomes positive, with no change to C's overall (zero) charge.
17A pair of horizontal parallel plates, 1.5 cm apart, is connected to a 900 V supply so that the top plate is positive.
a) Calculate the electric field strength between the plates. b) A small charged droplet of mass 4.9 × 10⁻¹⁵ kg is held stationary between the plates. Calculate the charge on the droplet. (g = 9.81 m s⁻²) c) State, with a reason, the sign of the charge on the droplet.
Hint: a) use E = V/d, converting the plate separation to metres. b) at balance, the electric force upward equals the weight downward, so qE = mg — rearrange to make q the subject. c) since the top plate is positive, the field between the plates points downward; for the electric force on the droplet to act upward (balancing weight), the droplet's charge must be negative.
18Two point charges, +3.0 µC and +3.0 µC, are fixed 40 cm apart, with a bar magnet placed well away from them on the bench.
a) Calculate the force between the two point charges. b) Describe the electric field line pattern you would expect around the pair of charges, and state the location of the point on the line joining them where the field strength is zero. c) Explain one similarity and one difference between the field line patterns of this pair of charges and the field around a single bar magnet.
Hint: a) substitute directly into Coulomb's law — both charges are positive, so the force is repulsive. b) field lines point away from each positive charge; by symmetry the zero-field point lies exactly midway between two equal charges. c) similarity: both sets of field lines never cross, and are closer together where the field is stronger. Difference: the electric field lines here start and end on the charges themselves, whereas a magnet's field lines always form closed loops, with no isolated "start" or "end" point.
Glossary
Charge (electric)
A property of matter, measured in coulombs, that causes objects to experience forces in electric fields.
Coulomb's law
F = kq₁q₂/r² — the equation for the force between two point charges.
Earthing (grounding)
Connecting an object to the ground so charge can flow freely, bringing the object to 0 V.
Elementary charge, e
The smallest possible unit of charge, 1.60 × 10⁻¹⁹ C; every charge is a whole-number multiple of e.
Electric field
A region of space in which a charge experiences an electric force.
Electric field strength, E
Force per unit charge, E = F/q, measured in N C⁻¹.
Field line
A line showing the direction of the force on a small positive test charge (or, for a magnetic field, the direction a compass would point) at each point; lines never cross, and are closer together where the field is stronger.
Induction (electrostatic)
Charge separation caused in an object by a nearby charge, without contact.
Magnetic pole
The north or south end of a magnet; like poles repel, unlike poles attract, and an isolated single pole has never been observed.
Point charge
A charge treated as existing at a single point in space.
Quantisation of charge
The fact that charge only ever occurs in whole-number multiples of e, as confirmed by Millikan's experiment.