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Topic 1: Electric Charge and Electric Fields

1.1 Charge

There is a “thing” called charge whose existence we deduce by observation. Charge, we observe;

• Comes in two type, called positive (+) and negative (-) which we sometimes call the sign.

• Comes in lumps, the smallest amount of which is the charge on an electron (which we call negative in

charge) and a proton (which is positive – the opposite to negative – in charge).

• Is conserved in the sense that the net amount of it cannot be change (charge can be created/destroyed, it is

just that equal amounts of positive and negative charges need to created/destroyed).

We need to associate a physical unit with charge – the SI unit is the coulomb and is given the symbol C. An

electron has a negative charge of magnitude 1.602 × 10-19 C. We will be concerned first of all for the case of

static charge (i.e. charge that is not moving).

1.2 Coulomb’s Law

We can do a bunch of experiments with charges and we find:

• Like charges repel and unlike charges attract – i.e. there is a force between charges which acts along the line

joining the charges and the direction of the force depends on the sign of the charges.

• The magnitude of the force between two charges is proportional to the magnitude of either of the charges

and inversely proportional to the square of the separation of the two charges.

• The experimental results can be summarised by Coulomb’s law,

221

4

1

r

QQF

πε= ,

where F is the magnitude of the force between the two charges Q1 and Q2 which are separated by a distance r.

The constant of proportionality is written asπε41 , where ε is the so-called permittivity of the medium in

which the experiment is done (the reason for writing the constant of proportionality in this way need not

concern us).

• For free space we write the permittivity as ε0 which has a measured value of 8.85 × 10-12 C2m-2N-1.

1.3 Atoms and Molecules

Atoms consist of charged particle:

• The nucleus consists of positively charges particles (protons) and neutral particles (neutrons) around which

move an equal number (to that of the protons) of electrons.

• The number of protons in a nucleus can vary from 1 to about 100.

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• Atoms behave in such a way that the electrons distribute themselves at different places (states) and one of

the consequences of this is that atoms can combine to form a wide range of molecules.

• Within a molecule electrons may distribute themselves “unevenly” amongst the atoms so that one part of

the molecule is more positive/negative than another part – such molecules are called polar and polar

molecules are chemically of great interest (e.g. water, DNA).

• In some arrangements electrons are mobile – and we have conductors, as opposed to insulators where the

electrons are not able to readily move. Intermediate cases are termed semiconductors which form an entire

branch of interest.

1.4 The Electric Field

There is a troublesome aspect of separated objects influences each other – the so called problem of action at a

distance. To “overcome” this problem we introduce the concept of a field and in the case of charge interaction

we introduce a so-called electric field.

• A charged particle has an electric field associated with it.

• Charged particles interact with (“feel”) electric fields.

• So charge particles interact via their electric fields.

• The electric field, E, is defined via the Coulomb force – imagine placing a positive charge, q, near a particle

of charge Q. then the force exerted on q by Q is,

24

1

r

QqF

πε= ,

so that dividing each side by q,

24

1

r

Q

q

F

πε= .

The left hand side is the force per unit charge that the charge q experiences – this is used to define the

electric field, E at the point where q is. Of course the charge q has its own electric field so that we need to

make q very small. So we define the electric field as,

q

FqE 0lim→=

• For a point charge Q the electric field is then,

24

1

r

QE

πε=

• For any arrangement of charge the electric field at any point is the superposition of the electric fields of

each of the charges at that point.

• Because force is a vector quantity so is the electric field

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1.5 Electric Field Lines

It is useful to represent electric fields pictorially by electric field lines which have the following properties:

• The electric field lines show the direction of the electric field – the electric field is tangential to the electric

field line

• The magnitude of the electric field is indicated by the density of the electric field lines

• Electric field lines originate on positive charge and terminate on negative charge – we put an arrow on field

lines to indicate the direction a positive charge place at rest would move.

• The electric field lines of combinations of charges can often be obtained by using the principles of

superposition.

1.6 Gauss’s Law

From just the idea of electric field lines we can deduce a good deal about the electric fields of a number of

arrangement of charges. The idea here is to consider the density of the electric field lines that pass through a

surface.

• Think about the density of the electric field lines that pass through the surface of a sphere which is centred on

a charge Q and recall that we said that the density of the field lines is a measure of the magnitude of the

electric field.

• Imagine now that we double the charge at the centre of the sphere – then we will also double the density of

lines through our sphere. So we can say that the electric field at the surface of (any) such sphere is,

QE ∝

• Now if we change the diameter of the sphere then the product of the density at the surface and the surface area

remains constant, so that,

=EA constant, k say

• The quantity EA is called the flux. Combining these last two expressions we have AQkE ∝ . With the result

of the electric field for a point charge than we see that ε1=k so that we arrive at Gauss’s law,

flux = εQ

EA =

• So Gauss’s law simply states that the electric field flux through a (closed) surface is equal to the charge,

divided by the permittivity, within the closed surface.

• We do not need to worry too much about this – we will use it once only to determine the electric field

between two oppositely charged plates.

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Lecture Problems

Charge

1. Calculate the magnitude of the force between two C-60.3 µ point charges 9.3 cm apart.

2. Calculate the repulsive electrical force between two protons m100.5 15−× apart from each other

in an atomic nucleus?

3. Two charged spheres are 8.45 cm apart. They are moved, and the force on each of them is

found to have been tripled. How far apart are they now?

4. Particles of charge 48,,75 ++ and C85µ− are placed in a line (Fig. 16–49). The center one is

0.35 m from each of the others. Calculate the net force on each charge due to the other two.

5. Two charges, 0Q− and ,3 0Q− are a distance l apart. These two charges are free to move but do

not because there is a third charge nearby. What must be the charge and placement of the third

charge for the first two to be in equilibrium?

Electric Field Lines

6. What are the magnitude and direction of the electric force on an electron in a uniform electric

field of strength CN2360 that points due east?

7. What is the magnitude of the acceleration experienced by an electron in an electric field of

?CN750 How does the direction of the acceleration depend on the direction of the field at that

point?

8. Draw, approximately, the electric field lines about two point charges, Q+ and ,3Q− which are a

distance l apart.

9. Determine the direction and magnitude of the electric field at the point P shown below. The

charges are separated by a distance 2a, and point P is a distance x from the midpoint between

the two charges. Express your answer in terms of Q, x, a, and k.

10. You are given two unknown point charges, 1Q and .2Q At a point on the line joining them,

one-third of the way from 1Q to ,2Q the electric field is zero (see below). What is the ratio

?21 QQ

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11. The two strands of the helix-shaped DNA molecule are held together by electrostatic forces as

shown in Fig. 16–44. Assume that the net average charge (due to electron sharing) indicated on

H and N atoms is 0.2e and on the indicated C and O atoms is 0.4e. Assume also that atoms on

each molecule are separated by m.100.1 10−× Estimate the net force between (a) a thymine and

an adenine; and (b) a cytosine and a guanine. For each bond (red dots) consider only the three

atoms in a line (two atoms on one molecule, one atom on the other). (c) Estimate the total force

for a DNA molecule containing 510 pairs of such molecules.

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EXPECTATIONS FOR TOPIC 1

ELECTRIC CHARGES AND ELECTRIC FIELD

By the end of this topic you should:

• Know what static electricity refers to

• Know what the conservation of charge means

• Know the difference between insulators and conductors

• Be able to use Coulomb’s law

• Know what an electric field is

• Know what electric field lines represent

• Have a “feel” for Gauss’s law (Gauss’s law and its consequences are not examinable)

You are not expected to:

• Be able to carry out two-dimensional vector addition

• Be concerned with electroscopes

• Know in any detail Gauss’s law

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PS123 Week 2 Tutorial Electric Charge and Electric Fields (Topic 1)

1. The form of Coulomb’s law is very similar to that for Newton’s law of universal gravitation. What

are the differences between these two laws? Compare gravitational mass with electric charge.

2. When determining an electric field, must we use a positive test charge, or would a negative one do

as well? Explain.

3. What is the total charge of all the electrons in 1.0 kg of H2O?

4. Two charges of equal magnitude exert an attractive force on each other of magnitude 4.56 × 10-3

N. If the charges are separated by 0.12 m what is the magnitude of the charges? Give your answer

in C, µC and nC. What can you say about the sign of the charges?

5. A proton is released in a uniform electric field, and it experiences an electric force of 3.75 × 10-14

N toward the south. What are the magnitude and direction of the electric field?

6. An electron is released from rest in a uniform electric field and accelerates to the north at a rate of

115 m/s2. What are the magnitude and direction of the electric field?

7. A pair of parallel plates with equal area of 1.23 ×10-6 m2 contain equal and opposite charges of

6.78 × 10-12 C. What is the electric field between the plates? By way of a diagram show the

direction of the field and indicated that the field is constant. Recall that E = σ/ε and σ = Q/A.