EM Section 1: Electric Charge

This section introduces the concept of electrical charge and provides a look at the properties and

importance of electric charge. Much of the section is revision but some parts are new.

1.1 Electric charge

The concept and definition of electrical charge

Humans have presumably known about certain electromagnetic phenomena such as lightning since they

first appeared on Earth. The first recorded experiments on producing electricity date back to the Greeks

over a thousand years ago when they discovered that rubbing an amber rod could cause it to attract light

objects like straw, and produce small sparks. By the 17th century William Gilbert was the first to start

classifying electrical and magnetic phenomena into distinct categories, and he coined the word “electric”

derived from the Greek word for amber.

A very loose definition of electric charge is simply a quantity that a body must have in order for it to

experience electric and magnetic phenomena of any kind. To be slightly more specific, charge can be

thought of as that quantity which is able to experience the effect of an electric or magnetic field. Electric

fields are covered in detail in section 3 and magnetic fields in section 8.

A better definition still, and the one that is most suited to this lecture course is that electric charge is that

quantity which causes a body to experience a force when placed in an electric field. Despite being

qualitative this is an unambiguous definition and will be used implicitly throughout the course.

The definition is similar to one of the ways of thinking about mass. Mass can be defined in several ways

but one useful one is that it is the quantity which causes a body to experience a force when placed in a

gravitational field.

Charge is a scalar quantity with units of coulombs ( C) and usually given the symbol 𝑞 or 𝑄 . The

magnitude of the charge on an electron is approximately 1.6 × 10−19 C and can be thought of as a

lower bound; a lightning strike might see a charge transfer between the cloud and the ground of the order

of 100 C. Most standard SI prefixes are used with the coulomb.

Unlike masses, charges can be either positive or negative. Of the two most important fundamental

charges, it is said by convention that electrons have a negative charge and protons have a positive charge

(which are of the same magnitude).

The net charge of the solid Earth is approximately −500 kC and the net charge of the whole Universe is

though to be zero viz. it is electrically neutral.

The force between static charges; Coulomb’s Law

As well as experiencing a force in an electric field, all charges act as a source of electric field. It therefore

follows that two neighbouring charges, 𝐴 and 𝐵, will both experience a force: charge 𝐴 produces a field

that causes charge 𝐵 to experience a force and charge 𝐵 produces a field that causes charge 𝐴 to

experience a force. By Newton’s third law, these forces must be equal in magnitude, of the same type

(electrostatic in this case) and act along the same axis.

In 1785, Charles Agustin de Coulomb, following on work by others, and following a series of careful

experiments, notably by using a torsion balance, deduced that:

(i) Any two charges, 𝑄1 and 𝑄2 put a force on each other of magnitude 𝐹 that is directly

proportional to the product of the two charges i.e. 𝐹 ∝ 𝑄1𝑄2

(ii) This force is inversely proportional to the square of the distance 𝑟 between the charges i.e. 𝐹 ∝

1

𝑟2

(iii) The forces is attractive when the charges have different signs and repulsive if the charges are of

the same sign

Combining these equalities and including a proportionality constant leads to Coulomb’s law of

electrostatics:

𝐹 = 𝑘𝑄1𝑄2

𝑟2 Equation 1.1a

The constant is found experimentally to be 𝑘 ≈ 8.9876 × 109 Nm2C−2. In SI units it is often written

as 𝑘 =1

4𝜋𝜖0 where 𝜖0 is known as the permittivity of free space. This is a more fundamental quantity than

the constant 𝑘 itself hence its rather clumsy looking inclusion in most texts on electromagnetism; its

importance will be revealed during this course and beyond. Permittivity in general is scalar quantity with

units of C2N−1m−2 = A2s4kg−1m−3, the latter being the SI base units, or, more usually, the Farad per

metre (Fm−1) and thus has a value of approximately 𝜖0 ≈ 8.8542 × 10−12 Fm−1.

Is it also worth noting now that 𝜖0 is essentially defined by 𝜖0 =107

4𝜋𝑐2 where 𝑐 is the speed of light in a

vacuum; this will be seen in section 9.

We can therefore write equation 11.a just as well as

𝐹 =1

4𝜋𝜖0 ∙

𝑄1𝑄2

𝑟2 Equation 1.1b

Usually in formal notes the lengthier version of equations incorporating 𝑘 will be used, but it is often

convenient to use 𝑘 when problem solving provided there is no danger of confusion with other terms.

The law can be written in vector format: if �̂� is a unit vector joining the two charges then the law

becomes

𝑭 =1

4𝜋𝜖0 ∙

𝑄1𝑄2

𝑟2 �̂� Equation 1.1c

and if 𝒓 is the displacement between the two charges then

𝑭 =1

4𝜋𝜖0 ∙

𝑄1𝑄2

𝑟3 𝒓 Equation 1.1d

The law is essentially the same as the inverse square law for gravity. Of course with gravity masses are

only always positive and the usual convention for the expression is to include a minus sign at the front to

indicate an attractive force. This isn’t necessary for electrostatics as the signs of the charges will give a

negative force (i.e. attraction) for unlike charges and a positive force for like charges.

The principle of superposition

Electrostatic forces add according to the principle of linear superposition. What this means is that if a

charge 𝑞 is in the vicinity of 𝑁 other discrete charges of different magnitudes and displacements from 𝑞,

the total force on 𝑞 is given by the vector sum of each of the other individual forces put on 𝑞 by the

other charges.

So if there are two neighbouring charges, 𝑞1 and 𝑞2 at displacements 𝒓1 and 𝒓2 then the force on 𝑞 due

to these charges is the vector sum 1

4𝜋𝜖0

𝑞𝑞1𝒓𝟏

𝑟13 +

1

4𝜋𝜖0

𝑞𝑞2𝒓𝟐

𝑟23 =

1

4𝜋𝜖0𝑞 (

𝑞1𝒓𝟏

𝑟13 +

𝑞2𝒓𝟐

𝑟23 ).

And if there are 𝑖 neighbouring charges the force on 𝑞 is given by 1

4𝜋𝜖0𝑞 ∑

𝑞𝑖𝒓𝑖

𝑟𝑖3𝑖

Rather than providing examples in the course notes problems involving Coulomb’s law and

superposition are at the end of the section and the first tutorial discussion problem.

Example: the force between a point mass and a radial line of charge

If two charges of value 𝑄 are separated by a distance 𝑑 they repel each other with a force given by

𝐹 = (1

4𝜋𝜖0)

𝑄2

𝑑2.

If one of the charge were rolled out into a uniform line of charge of length 𝐿 and uniform charge

density 𝜆 =𝑄

𝐿 pointing at the point charge with the centre of the line at distance 𝑑 as shown below will

the force increase, decrease or stay the same?

One can argue that the force should be greater: the half that is closer is going to see greater increase in

force than the half that is further away by the inverse square law. In the limit that the spread out charge

overlaps with the point charge it would seem that the force should become infinite further justifying the

guess.

Herewith the detailed calculation:

Consider the force between the point charge and small element of the line of charge of width 𝛿𝑥 a

distance 𝑥 from the point:

The element has a charge 𝜆. 𝛿𝑥 =𝑄

𝐿𝛿𝑥 and force between the point charge and the element is, by

Coulomb’s law, 𝛿𝐹 = (1

4𝜋𝜖0)

𝑄.𝑄

𝐿𝛿𝑥

𝑥2 = (1

4𝜋𝜖0)

𝑄2

𝐿

𝛿𝑥

𝑥2.

The total force is, by the principal of superposition, given by the sum of each element of the line in the

limit that 𝛿𝑥 → 0 and the number of terms in the sum becomes infinite. In this limit the sum is an

integral given by

𝐹 = ∫ (1

4𝜋𝜖0)

𝑄2

𝐿

𝑑𝑥

𝑥2

𝑥=𝑑+𝐿

2

𝑥=𝑑−𝐿

2

The limits on the integral correspond to the ends of the rod.

As 𝑥 is the only variable in the integral we can write

𝐹 = (1

4𝜋𝜖0)

𝑄2

𝐿∫

𝑑𝑥

𝑥2

𝑥=𝑑+𝐿

2

𝑥=𝑑−𝐿

2

which is easily integrated to give

𝐹 = (1

4𝜋𝜖0)

𝑄2

𝐿[−

1

𝑥]

𝑥=𝑑−𝐿

2

𝑥=𝑑+𝐿

2

This now becomes 𝐹 = (1

4𝜋𝜖0)

𝑄2

𝐿{−

1

𝑑+𝐿

2

+1

𝑑−𝐿

2

} = (1

4𝜋𝜖0)

𝑄2

𝐿{

1

𝑑−𝐿

2

−1

𝑑+𝐿

2

} = (1

4𝜋𝜖0)

𝑄2

𝐿{

𝑑+𝐿

2−(𝑑−

𝐿

2)

(𝑑−𝐿

2)(𝑑+

𝐿

2)}.

Some happy cancellation and simplification happens now; the numerator of the part in curly brackets

becomes 𝐿 and the denominator 𝑑2 −𝐿2

4=

1

4(4𝑑2 − 𝐿2).

The force is thus 𝐹 = (1

4𝜋𝜖0)

𝑄2

𝐿∙

𝐿1

4(4𝑑2−𝐿2)

which simplifies to a final answer of

𝐹 = (1

4𝜋𝜖0)

4𝑄2

(4𝑑2 − 𝐿2)

Taking the length of the rolled out charge as the only variable here means we have a function of the

form 𝑦 =1

1−𝑥2 and plot of 𝐹 vs. 𝐿 is shown below:

This corroborates the guess that the magnitude of the force rises, confirms it becomes infinite when the

charges overlap (i.e. when 𝐿 = 2𝑑) and reduces to the force between point charges when 𝐿 = 0 as it

should do.

Point charges

Coulomb’s law of electrostatics applies to point charges only i.e. charges of infinitesimally small size

concentrated at one point in space - a hypothetical idealisation that can never really exist. But in terms of

their electrostatic effect any macroscopic charge can usually be thought of as having its net charge acting

from one point. For uniformly charged spheres the electrostatic effect is equivalent to that of the net

charge on the sphere acting as a point mass located in the centre. This is studied in detail in section 3 on

Gauss’s law.

The conservation of charge

Charge is a conserved quantity. The principle of the conservation of charge states that the total

charge of an isolated system is constant.

This principle is always true – there are no known exceptions.

So for example, in free neutron decay, a neutron splits into a proton and an electron, thereby meaning

the net charge before and after decay is zero:

𝑛 → 𝑝+ + 𝑒− + 𝜈�̅�

Of course, as well as conserving charge, the decay process also conserves mass-energy, linear

momentum and angular momentum as should already be familiar. The third product on the right hand

side is an antineutrino which satisfies a third conservation law: conservation of lepton number; this is

covered in year 2 Nuclear and Particle Physics.

The quantisation of charge and the electron

In 1909 a classic experiment by Robert Millikan suggested that charge exists in discrete units, or quanta,

of magnitude 𝑒 ≈ 1.6021 × 10−19 C.

Subsequent experiments have never shown an isolated charge to have a lower value and thus this can be

though of as the fundamental quantum of charge. It is now well known that this is the magnitude of the

charge on a single electron (which is negative) and on a proton (which is positive). Note that the electron

is not the charge in itself – it is merely a particle, with several properties, one of which is its electrical

charge.

Millikan’s experiment will not be covered in detail in this course but will be referred to occasionally. It is

one of the experiments in year 2 laboratory.

The electron is thought of as a fundamental particle, but the proton is now thought be composed of

three quarks. The most well accepted theories on the nature of quarks suggests they have fractional

charges with values of ∓1

3𝑒 and ∓

2

3𝑒. However, as a quark has never been isolated and they only ever

seem to exist in bound states of two or more with charges ≥ 𝑒 it can still be said that 𝑒 is the quantum of

isolated charge. You may also see electrostatics problems in textbooks requiring computation of the

forces between quarks within a proton; these questions should not be taken too seriously as there is no

evidence that quarks do experience electrostatic forces in this way.

Free and bound charges

If an electron is not confined within an atom, ion, or other microscopic assembly of particles then it is

said to be a free electron. For many purposes, free electrons and other free particles can be treated using

classical mechanics. This essentially means their behaviour when under the influence of gravitational,

magnetic and electric fields can be treated using the relevant force field laws with 𝐹 = 𝑚𝑎.

But if an electron is confined with an atom then the electron is said to be a bound electron. The

behaviour of a bound electron or any other bound particle is governed by quantum mechanics, in

particular the Schrödinger equation. Applying classical mechanics and 𝐹 = 𝑚𝑎 to such objects is not

good physics and should be avoided.

One remarkable consequence of quantum mechanics is that in most metallic solids and some other

conductors and semiconductors, each atom contributes one conduction electron to the material. Though

these electrons are part of the metal (though can be easily removed) they can, to a certain extent, behave

classically as free electron within the metal. That this happens to be true is far from obvious; it is covered

in year 2 Solid State physics following two in-depth courses on quantum mechanics.

Types of charged particle

There are several types of charged particle. The most common, and the ones that will be used most

often in this course are electrons, protons and ions. Though you won’t meet them formally at university

until Solid State physics in year 2 it is also worth knowing that in semiconductor devices, an absence of an

electron in a lattice is known as “hole” and acts as a positive charged particle of charge +𝑒. It is not the

same as a positron, which is an antimatter particle.

Positrons are just one of a myriad of other fundamental charged particles that you will start to study

formally in year 2 Nuclear and Particle physics but will not be of much relevance in this course.

Charge density

Charge density is defined as the amount of charge per unit of space. It can be defined in one, two or

three dimensions:

• Linear charge density is the amount of electric charge per unit length in Cm−1. It is often given

the symbol 𝜆.

• Surface charge density is the amount of electric charge per unit of surface area in Cm−2. It is

often given the symbol 𝜎.

• Volume charge density is the amount of electric charge per unit of surface area in Cm−3. It is

often given the symbol 𝜌.

1.2 Movement of charge and conduction

Electric current

When electric charges move relative to a fixed reference frame then there is said to be an electric

current. The electric current, 𝐼, through a surface is defined as the rate of flow of charge through the

surface. This is expressed mathematically by

𝐼 =𝑑𝑄

𝑑𝑡 Equation 1.2

Electric current is a scalar quantity with units of Cs−1 or amperes or amps (A). The electric currents in

human nerve endings are of the order of picoamps; a lighting strike might have a current as high as

several kiloamps. All standard SI prefixes are commonly used with the ampere.

Equation 1.2 is for instantaneous current. For the average current over a relatively long time the related

equation is

𝐼𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =∆𝑄

∆𝑡 Equation 1.3

A positive current can be given either by the flow in one direction of positive charge, or in the opposite

direction to a flow of negative charge. As electrons are the main charge carriers in conductors this is

usually the case. When a wire carries a positive current it corresponds to a flow of electrons in the

opposite direction. If there are two signs of charge carriers, for example when a potential difference is put

across an ionised fluid, then both positive and negative charged carriers contribute to the total current.

Flow of charge is covered in more detail in section 6.

The ampere

When defining a quantity in physics it is usually useful to clarify

(i) whether it is a scalar or vector

(ii) its SI unit, its SI base units, and any other units in common usage it might have

(iii) what range of values the quantity takes, including whether there are upper and lower bounds to

the value

(iv) the corresponding definition of one single SI unit of the quantity.

In the SI system of units the base units in classical mechanics are the metre (for length), the second (for

time) and the kilogram (for mass). Then in thermodynamics there is the kelvin (for temperature), when

dealing with matter the mole (for amount of substance can be used) and in studies on light the candela

(for luminous intensity) can also be used.

The ampere is the SI unit that appears in electromagnetism, and most base SI units that appear within

the subjects are multiples of amps, seconds, kilograms and metres (as with the base units for permittivity

for example).

The base SI units are defined in rigorous fashion and the definitions can be somewhat complicated as

this is necessary to remove any ambiguity. The ampere is no exception. It is definition that you will be

expected to know, but as it involves a discussion of magnetic fields and the forces between current

carrying wires, this is left until section 8.

The coulomb

Though charge and its SI unit have been defined, the coulomb itself has not been defined. Its definition

is quite simple and follows from the definition of electric current: one coulomb is defined as the charge

that passes through a surface when a constant current of one ampere flows for one second.

1.3 Charging and discharging

It is worthwhile going over some of the simple physics involved with charging and discharging

macroscopic objects in electrostatics:

Charging by friction

When two different materials rub together there is often a transfer of electrons from one object to the

other. One of the materials has a greater affinity for electrons than the other so the rubbing of the two

objects together causes electrons to move from one material to the other leaving one material with a

positive charge (caused by a loss of electrons) and the other with a negative charge. This is called charging

by friction.

For example, the rubber of a balloon is more electronegative than human hair, so rubbing a balloon on

you chair causes the balloon to end with net negative charge and your hair to end with a net positive

charge.

Charging by induction

Electrostatic induction is the redistribution of charge in one object caused by the movement of another

charged object. This can be used to create a net electrical charge on a previously neutral object. There are

several ways in which this can be done but what follows is one of the simplest.

Consider two touching metal spheres as shown in figure 1.3a:

Figure 1.3a Two electrically neutral conducting spheres in contact with each other

Now consider what happens if a positively charged object is brought close to one side of the two

spheres. As the spheres can conduct they have free electrons that respond to an electric field. This means

electrons in the conducting spheres will move towards the positive charge. This creates a charge

distribution with the sphere nearer the external positive charge having a negative charge as it has an

excess of electrons and the sphere further away having a positive charge as it has a deficit of electrons.

This is shown in figure 1.3b:

Figure 1.3b Charge redistribution on the spheres

If the positively charged sphere is now moved away it retains its net positive charge, and if the external

positive is moved away then the other sphere retains its net negative charge. The two spheres have been

charged by induction and their charges are equal in magnitude and opposite in sign.

Charging by induction can also be done on single conductor by bringing the external charge close by

and the simply touching the other side of the sphere to connect it to ground. Electrons flow so as to

neutralise the side being touched leaving the sphere with an overall charge.

Charging by contact (also known as charging by conduction)

It is also possible to charge a conducting object by bringing another charged object into contact with it.

This causes electrons to flow from one object to the other to even out the charge distribution. The

previously charged object finished with less of a net charge and the previously uncharged object gains the

loss.

Discharging

It is easy to remove the charge from an object simply by touching it. This is known as “grounding” or

“earthing”. You are an electrical conductor so electrons which are in an infinite supply in the earth at your

feet will always flow to electrically neutralise any object being touched.