Fields Fields of force Gravitational fields Circular motion in the solar system – satellites Electric fields
Electric potential Magnetic fields Applications of electric and magnetic fields Electromagnetic
induction
First folder section, first page
Fields of force
Fields of force provide a way to explain action at a distance: the fact that two objects can have an
effect on each other without being in contact. Different objects are affected by different kinds of field.
For example, any object which has a mass will feel the effect of a gravitational field, and in turn will
create a gravitational field of its own. The three fields we will be dealing with (and the type of objects
they relate to) are shown in the table.
Field
Gravitational
Electric
Magnetic
Object characteristic
Mass
Charge
Moving charge or magnet
We recognize a field through its effects: we can deduce that an electric field is present in a region if
charged objects experience a force. This is also how we measure fields.
Drawing fields of force
A force is a vector quantity, and so is a force field. Remember that a vector has both a magnitude and a
direction. We can represent a force field by drawing lines, using the following rules:
 The density of the lines indicates the strength of the field – this means that when the lines are
closely-spaced, the force is strong
 The lines are labelled with arrows to show the direction of the field.
Field strength
We can study a field through its effects. To measure the field strength, we must place a suitable object
in the field and measure the force on the object. A stronger field produces a larger force. “Suitable”
means two things:
 The right kind of object must be used! For example, a charged object is needed to measure an
electric field.
 The object must be small, so that it does not alter the field being measured.
Uniform fields
A uniform field has a constant strength and direction. We represent it using equally-spaced parallel
lines:
An object placed in a uniform field experiences the same force everywhere.
Non-uniform fields
To represent a non-uniform field, we use the rules above. The lines must get closer together as the field
strength increases.
Change in potential energy in fields of force
The presence of fields may make it harder or easier to move objects around. For example, the presence
of the earth’s gravitational field makes it much easier to fall down than to jump up! This can be
described in terms of energy.
 When you move an object against a field, you have to do work. This means you must supply
energy to the object, and the potential energy of the object increases.
 When you move an object in the direction of a field, the object does the work! Energy is
released by the object, and its potential energy decreases.
When you lift a book in the air, you do work, and the potential energy of the book increases. If you
drop the book, it will fall, and the potential energy decreases again – in this case, it is converted to
kinetic energy.
The change in potential energy is equal to the work done in moving the object. It is positive when
moving against the field, and negative when moving with the field.
Change in potential energy when moving at an angle to a field
It is still possible to calculate the change in potential energy when an object moves at an angle to the
field line. We must still calculate the work done in moving the object. However, to do this, we now use
the component of the force along the direction of motion (see Mechanics notes):
Force in the direction of motion = F cos θ
Absolute potential at a point in a field
We have seen that we may calculate the change in potential energy of an object as it is moved around
in a field. We would like to be able to give an absolute value to this energy (so that we can say: “The
potential at this point is 10 J”).
The solution is to choose a zero of potential, and the normal choice is to let the potential energy be zero
at infinity. The potential at a point X is then the work done in moving an object from infinity to X.
Letting the potential be zero at infinity is not the only choice, and is not always the best one. When
dealing with problems involving motion in the earth’s gravitational field close to the earth’s surface, it
is usually more convenient to let the zero of potential be at the surface. We then say that an object on
the surface has zero potential energy, and calculate changes from there.
Equipotential lines
By joining together all points at the same potential in a field, we create equipotential lines.
Notice that the equipotential lines always cross the field lines at right angles. This is because when we
move an object in a direction perpendicular to the field, we don’t need to do any work – and the
potential energy does not change.
Second folder section, first page
Gravitational fields
Any object with a mass experiences a force when placed in a gravitational field. This is how
gravitational fields may be detected. The other key characteristics of gravitational fields are:
 They are generated by objects which have mass
 The force is always attractive
 The force is weak in comparison with other fields.
The gravitational field strength at a point is defined as the force per unit mass at that point. It is
labelled g and has units of N kg-1. To measure the field strength accurately, a small test mass must be
used – the larger the mass, the more the original field is altered.
The lines of force around a spherical body always point inward. This is because the gravitational force
is always attractive.
Notice that the lines of force are denser closer to the mass M. This shows that the field becomes
stronger here.
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Newton’s law of gravitation
Newton’s law of gravitation tells us the strength of the gravitational force between two objects. It
applies to point masses (idealized objects where all the mass is concentrated at a point). The force
between two masses m1 and m2 separated by a distance d is
F
Gm1m2
d2
G is the gravitational constant. It has the value 6.67 × 10 -11 N m2 kg-2.
This law may also be applied to spheres of uniform density. We have to treat the spheres as though all
the mass were concentrated at the centre. Note that this does not work inside a sphere!
The gravitational constant, G, is a very small number. This is because the gravitational force is weak.
However, because the force is always attractive, it is still possible for it to have a strong effect – but a
lot of mass is required! An example of “a lot of mass” is 6.0 × 10 24 kg – the mass of the earth, ME.
The force is proportional to the inverse of the distance squared – the law of gravitation is an inverse
square law.
Remember that the force is the same size for both masses, but in the opposite direction (since it is
attractive). We could have predicted this from Newton’s third law of motion (see the Mechanics notes
again!).
Gravitational field strength and Newton’s law
We can use the information we already have to find the gravitational field strength due to a point mass
M. Remember that the field strength is defined in terms of the force F on a mass m placed in the field:
g
F
m
Using Newton’s law, we can calculate this force F, which then allows us to find g.
GMm
d2
GMm GM
g
 2
md 2
d
F
A gravitational field of great importance to us is the one produced by the Earth. We can calculate the
strength of this field. On the surface, the distance from the centre is rE (the radius of the Earth). The
gravitational field strength at the surface is
g
F GM E m GM E

 2
m
mrE2
rE
New page
The acceleration due to gravity and the gravitational field
strength
Like any other force, the gravitational force causes acceleration. Newton’s second law of motion says
that
F = ma
We can calculate the size of the acceleration on the surface of the Earth. For a mass m, the gravitational
force on the Earth’s surface is
F
GmM E
rE2
If we divide by m, we get the acceleration due to gravity on the Earth’s surface:
a


GM E
rE2
Note that this expression does not involve m (the mass of our object). This tells us that all
objects fall at the same rate (ignoring air resistance)!
Compare this expression with the gravitational field at the surface. They are the same! These
two different quantities have the same numerical value:
Acceleration due to gravity (9.81 m s-2) = Gravitational field strength (9.81 N kg-1)
Remember that this only applies at the Earth’s surface. On the surface of the Moon, for example, the
value is much lower (1.62 m s-2).
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Mass and weight
These two quantities are often confused, but they are very different.
 Mass is a property of an object, which tells us how it will respond to an external force. If we
apply the same force to two bodies, the one with the greater mass will accelerate more slowly.
This is just a restatement of Newton’s second law of motion (F = ma). Mass is measured in
kg.
 Weight is the force on an object in a gravitational field. If the gravitational field strength is g,
an object with mass m has weight F = mg. Note that the weight of an object changes with the
strength of the gravitational field. You would weigh less if you were standing on top of Mount
Everest – and even less on the surface of the Moon. Weight, being a force, is measured in N.
It is incorrect to say “My weight is 70 kg”! More accurate would be “My weight on the Earth’s surface
is 687 N” or “My mass is 70 kg”.
Third folder section, first page
Circular motion in the solar system
When an object moves in a circle, it is constantly accelerating towards the centre. (Although its speed
may be constant, its velocity certainly isn’t – see Mechanics notes.) The force which produces this
acceleration is called the centripetal force.
The centripetal force which keeps the Earth orbiting around the Sun is provided by gravity.
The gravitational force between the Sun and the Earth is
F
GM E M S
R2
This must be equal to the centripetal force required to keep the Earth in circular motion with speed v,
which is
F
M Ev2
R
We can now combine these expressions to find out how fast the Earth is moving.
GM S M E M E v 2

R2
R
GM S
v
R
This speed does not depend on the mass of the Earth! Any object orbiting the Sun at the same distance
travels at the same speed.
Kepler’s law
Kepler’s law relates the radius and the period of the orbit. The period T is related to the speed:
v
2R
T
We can now derive Kepler’s law using the equations of the previous section.
GM S M E M E v 2

R2
R
4 2 R 2 M E

T 2R
4 2 R 3
T2 
GM S
T 2  R3
Kepler’s law states that the square of the period of rotation is proportional to the cube of the orbital
radius.
Satellites
Our analysis of the motion of the Earth around the Sun can equally be applied to the motion of a
satellite around the Earth. We replace the Sun’s mass with the Earth’s and obtain the orbital speed:
v
GM E
R
This only depends on the orbital radius, since G and ME are constants. If we are told that a satellite is
orbiting at a height h above the Earth’s surface, the orbital radius is
R  h  rE
We are able to choose the speed of the satellite’s orbit simply by varying the orbital radius. A larger
radius means a slower satellite!
This allows us to put satellites in geostationary orbit. If we choose the speed so that the orbital period
is one day, and set the satellite orbiting in the plane of the equator, the satellite always remains above
the same place on the Earth’s surface. This is very useful for communication satellites.
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Gravitational potential
When we move an object with mass against a gravitational field, we have to do work (think about
carrying a heavy object upstairs). The work we do is stored as potential energy by the object. This
shows that there is a potential energy associated with any object which has mass in a gravitational field.
In the diagram, moving the mass m away from mass M requires us to do work, so this brings an
increase in potential energy. Energy is released as mass m moves towards mass M, so this means the
potential energy decreases.
We define the gravitational potential at a point in a gravitational field as the work done in bringing a 1
kg mass from infinity to the point.
It is possible to use this definition, combined with Newton’s law of gravitation, to calculate the
potential at a distance R from an object of mass M:
V 
GM
R
The gravitational potential is always negative. This is because our definition means that V is zero at
infinity, and because the gravitational force is always attractive.
Remember that V is the potential energy for a mass of 1 kg. The potential energy of a mass m is mV.
Gravitational field strength and potential gradient
A mass m is in a gravitational field, at a point where the field strength is g. If I move the mass a small
distance Δx, the work done is
W   Fx
 (mg )x
The minus sign is required because if I move the mass with the field, ΔW is negative. The change in
potential energy of the mass is
mV  W
Combining these two equations tells us that
 (mg )x  mV
V
g
x
This is the potential gradient. We have shown that the gravitational field strength is numerically equal
to the potential gradient.
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Potential and kinetic energy of satellites
The potential due to the Earth’s gravitational field at a distance R from the Earth’s centre is
V 
GM E
R
This means that a satellite of mass m in this position has potential energy
EP  
The kinetic energy is E K 
GM E m
R
1 2
mv , which we can rewrite using our expression for v:
2
EK 
GM E m
2R
The total energy of the satellite is the sum of the potential and kinetic energies:
E  EP  EK

GM E m
2R
As the satellite comes closer to Earth, the kinetic energy increases, but the potential energy decreases.
The potential energy decrease ‘wins’ – the energy of the satellite decreases as it comes closer to Earth.
This means that in order to move into a lower orbit, the satellite must lose energy. It also explains why
there is a concern about satellites falling out of the sky! Any frictional force, however small, will cause
the satellite to lose energy, and so to move into a lower orbit, and eventually to fall back to Earth.
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Escape velocity
Imagine launching an object vertically into the sky. How fast would the object have to be travelling in
order to escape from the Earth’s gravitational field? The answer is the escape velocity. At this speed,
the object has just enough kinetic energy to avoid falling back to Earth.
An object of mass m on the surface of the Earth has gravitational potential energy

GM E m
rE
When the object escapes from the field, the potential energy is zero. The change in potential energy is
therefore
GM E m
rE
This is the amount of kinetic energy which must be supplied to the object when it is launched. Equating
these two quantities lets us work out the escape velocity vE:
1 2 GM E m
mvE 
2
rE
vE 
2GM E
rE
 2 grE
Note that (once again) this does not depend on the mass of the object.
Fouth folder section, first page
Electric fields
Electrical charge, like mass, is a property of an object. However, unlike mass, electrical charge may be
positive or negative.
Any charged object experiences a force when placed in an electric field. This is how electric fields may
be detected. The other key characteristics of electric fields are:
 They are generated by charged objects
 The force may be attractive or repulsive
 They are strong fields.
Like charges repel each other, whereas unlike charges attract:
The electric field strength at a point is defined as the force per unit positive charge at that point. It is
labelled E and has units of N C-1. The letter C stands for coulomb, which is the unit of charge. It is
important to remember that the direction of the field is the direction of the force on a positive charge.
The force on a negative charge is in the opposite direction.
E
F
Q
Q is the letter normally used to represent charge.
To measure the field strength accurately, a small test charge must be used – the greater the charge, the
more the original field is altered.
We can rearrange the equation defining the electric field strength to tell us the force on a charge Q in
an electric field of strength E:
F  EQ
Remember that F and E are both vectors – they have a magnitude and a direction.
Uniform electric fields
The simplest kind of electric field is a uniform field.
One way of generating a uniform field is to have two oppositely-charged parallel plates. The field lines
go from the positive plate to the negative plate.
When a charged particle passes between the plates, it is deflected. Note that the negative charge is
deflected towards the positive plates.
This is how oscilloscopes work. A stream of charged particles is fired at a screen, passing through a
pair of parallel plates on the way. The amount of charge on the plates affects the strength of the field,
which in turn affects how much the particles are deflected. The amount of deflection visible on the
screen therefore gives information about how much charge is on the plates.
Coulomb’s Law
The equivalent of Newton’s law of gravitation for electric fields is Coulomb’s law. It tells us the force
between two charges, Q1 and Q2, separated by a distance d.
F
1
Q1Q2
40 d 2
The constant ε0 is called the permittivity of free space and has the value 8.85 × 10 -12 C2 N-1 m-2.
Remember that the force is attractive when Q1 and Q2 have opposite signs, but repulsive when they
have the same sign.
Like Newton’s law of gravitation, this is an inverse square law. This is something the electric and
gravitational fields have in common.
Electric field due to a point charge
We measure the electric field through its effect on a test charge. If we place a test charge q a distance d
away from a point charge Q and measure the force, we can then deduce the electric field strength:
This tells us the field strength at one point in space. If we repeat the measurement for lots of different
points, we can plot the electric field of a point charge:
These are non-uniform fields. The field lines radiate outwards for the positive charge, but point inwards
for the negative charge. The arrows show the direction of the force on a positive charge – a positive
charge placed in the field of another positive charge will be repelled, while a positive charge placed in
the field of a negative charge will be attracted. This means that electric field lines must always start at
positive charges and end at negative charges.
The density of the field lines is greater near to the centre, because the field is stronger there.
Other non-uniform fields
An electric dipole consists of two charges of the same magnitude but opposite sign. The field lines for a
dipole go from the positive charge to the negative charge.
Outside the hollow sphere, the field looks just like that of a point charge (in this case, a positive point
charge). However, there is no electric field inside the sphere. This can be very useful if you want to
protect something from external electric fields. A surprising example is the body of an aircraft. Aircraft
are quite frequently struck by lightning, but the metal body of the aircraft protects those inside from the
harmful effects of the strong electric field.
The electric field created by a flat conducting sheet is perpendicular to the surface:
If the charge per unit area of the sheet is σ then the magnitude of the field is
E

0
5th folder section, first page
Electric potential
Charges placed in an electric field have potential energy. To see this, think about pushing two positive
charges closer together. This requires work, and the energy supplied to do it is stored by the charges as
potential energy, which may be released if the charges are allowed to separate.
As with gravitational fields, it is possible to define an absolute potential. We define the electric
potential at a point as the work done per unit charge against the electric field in bringing the charge
from infinity to that point.
V
W
Q
V is the electric potential at a point; W is the work done in bringing the charge Q from infinity to that
point.
The potential is measured in volts (V). We know that W is measured in joules (J), and the charge Q is
measured in coulombs (C). This means that 1 V = 1 J C-1.
The electric potential may be positive or negative. This is because of the existence of positive and
negative charges and the fact that the electric force may be attractive or repulsive. This is in contrast to
the gravitational field, which is always attractive, and the gravitational potential, which is always
negative.


To determine whether the potential at a point is positive or negative, ask the question: “Do I
have to do work when I move a positive charge from infinity to this point?”
If the answer is “Yes”, the potential is positive, otherwise the potential is negative (unless the
point is at infinity, in which case the potential is zero!).
Potential due to a point charge
The electric field created by a point charge is in the radial direction (either outward or inward).
For the potential, what matters is the distance from the charge. All points which are the same distance
from the charge must have the same potential. This means that the equipotential surfaces are concentric
spheres, centred on the charge.
Potential difference



The potential difference between two points is the change in potential energy when a positive
charge moves from one point to the other.
This is equivalent to the work done (per unit positive charge) when moving a charge from one
point to the other.
The potential difference does not depend on the route you take.
This potential difference is the same one you encountered when studying electrical circuits.
The electric field and potential gradient
A charge Q is in an electric field, at a point where the field strength is E.
The force is F  QE . If I move the charge a small distance Δx, the work done is
W   Fx
 (QE )x
The minus sign is required because if I move the charge with the field, ΔW is negative. The change in
potential energy of the charge is
QV  W
Combining these two equations tells us that
 (QE )x  QV
E
V
x
This is the potential gradient.


We have shown that the magnitude of the electric field equals the potential gradient
The minus sign shows that the electric field is in the direction of decreasing potential.
We found the same result for the gravitational field. The electric field strength E is the electric
equivalent of the gravitational field strength g.
In a uniform field, the electric field and the potential gradient are constant. For the common system of
two plates with a potential difference V separated by a distance d, the electric field is
E
V
d
Remember that the electric field lines go from positive to negative charge.
Equipotential lines
Remember that equipotential lines are always at right angles to the field.
The equipotential lines for a uniform field are straight, and perpendicular to the field lines. The
equipotential lines for a non-uniform field are more complicated.
We saw previously that the equipotential surfaces for a point charge are concentric spheres. Since the
field lines are radial, they are at right angles to the equipotential surfaces, as we expect.
6th folder section, first page
Magnetic fields
Magnetic fields, like electric fields, are produced and detected by charged objects. However, there is a
difference – magnetic fields are associated with moving charges.
Two familiar examples of objects which generate magnetic fields are:
1. Current-carrying wires. The moving charges are the electrons in the wire.
2. Magnets! The moving charges are electrons orbiting the atoms in the magnet.
In fact, every atom has moving charges, and produces a magnetic field like a tiny bar magnet. Magnetic
materials are special because all the atomic magnetic fields line up, rather than pointing in random
directions and cancelling each other out.
We can use the simple bar magnet to define the direction of our magnetic field:
The field lines go from the north pole to the south pole. The north pole of the magnet is the one that is
attracted to the north pole of the Earth.
The simplest magnetic objects (bar magnets or current loops) have two poles. There are no magnetic
monopoles – it is not possible to have a north or south pole in isolation. If you cut a bar magnet in half,
you are left with two smaller bar magnets, each with two poles! This is an important difference
between magnetic fields and electric/gravitational fields.
When a current-carrying wire is placed at right angles to a magnetic field, it experiences a force. We
use this force to define the magnetic field strength, B. Also known as the magnetic flux density, it is the
force acting per unit current per unit length. In symbols,
B
F
IL
I is the current, L is the length of the wire, and F is the force as usual. Magnetic field strength is
measured in tesla (T).
Remember that the wire must be at right angles to the field. The force on the wire is then at right angles
to both the wire and the field.
“Current” means conventional current – the direction of positive charge flow.


The force is always at right angles to both the field and the current
If the field and the current are in the same direction, there is no force!
If we want to measure a magnetic field, we need to measure the force on a test object – a small current
loop or bar magnet. As for the electric and gravitational fields, we use a small object to avoid
disturbing the field.
Magnetic flux
When we have a magnetic field, we can choose an area A and define a new quantity – the magnetic
flux, Φ.
The magnetic flux is the magnetic field strength multiplied by the area.
  AB
This tells us “how much” magnetic field is passing through a region. It is measured in weber (Wb).
Note that the area must be perpendicular to the field.
Magnetic field patterns
It is important to remember the direction of the field produced by a straight wire. You can do this using
the right-hand rule (also known as the screw rule). Make a “thumbs-up” sign with your right hand, and
point your thumb in the direction of the current. Your fingers then curl around in the direction of the
field. Compare it with the field diagram to make sure you’re doing it properly!
The solenoid is easy once you’ve learnt the current loop (it’s just lots of them in a row). When the
current is in the clockwise direction, the field inside the loop is pointing into the paper.
New page
Force produced by a current in a magnetic field
By rearranging our definition of the magnetic field strength, we can calculate the force on a wire in a
magnetic field.
These equations give us the force on a length L of a long wire. Note that we have now relaxed the
condition that the wire must be perpendicular to the field.
We can see what happens for different values of θ:
 When θ = 90˚, the wire is perpendicular to the field and feels the full force BIL.
 When θ = 0˚, the wire and the field are in the same direction and the force is zero.
 In between, the force rises from zero to the maximum value BIL.
Forces on coils
We can use the force on a straight wire to calculate the force on a coil. Each turn of the coil is a
rectangle.
In the diagram, sides b→c and d→a are in the direction of the field, so the force they experience is
zero. Sides a→b and c→d are perpendicular to the field, and both experience a force of magnitude BIL.
However, because the current is flowing in opposite directions in a-b and c-d, the forces are also in
opposite directions. The result is a torque or turning force on the coil of magnitude BILd.
This arrangement forms a simple motor, converting electrical current into rotation with the help of a
magnetic field.
Forces on moving charges
Current is made up of moving charges. In mathematical terms, the current at a point is
I
Q
t
Q is the total charge flowing past the point in time t. We can use this to help us derive a formula for the
force on a moving charge in a magnetic field.
If Q is the total charge in a length L of the wire, then in time t the charge will have travelled a distance
L  vt
v is the speed at which the charge is travelling.
We can substitute these results into our formula for the force on a current-carrying wire:
F  BIL
Q
L
t
 BQv
B
This is the force on a moving charge in a magnetic field.
To find the direction of the force, we may continue to use our left-hand rule (Motion, Field, Current),
but we must remember that the current is only in the same direction as the moving charge when the
charge is positive. For negative charges, we treat the current as being in the opposite direction.
The force is always at right angles to the direction of movement. This is the condition for circular
motion! A moving charge in a magnetic field will travel in a circle. The magnetic field creates the
centripetal force.
In the diagram, the field is pointing into the page. The magnetic force is
F  Bqv
A mass m moving at speed v in a circle of radius r requires a centripetal force
F
mv2
r
This is provided by the magnetic force, so the two expressions must be equal.
mv2
Bqv 
r
mv
r
Bq
The radius of the motion in a given field is determined by the speed, the mass, and the charge. This has
many useful applications, some of which will be discussed next.
7th folder section, first page
Applications of electric and magnetic fields
We now have enough information to understand the behaviour of charged particles in electric and
magnetic fields. There are many examples of the application of these fields to moving charges.
Using an electric field to accelerate a charged particle
An electric field may be used to increase the velocity of a charged particle. Electric potential energy is
converted to kinetic energy as the velocity is increased.
The force on the positive charge is towards the negative plate, while the force on the negative charge is
towards the positive plate.
If the plates are separated by a distance d, the electric field strength has magnitude
E
V
d
From this, we can calculate the force on the charged particle, and hence the acceleration (from
Newton’s second law of motion):
Vq
d
F Vq
a 
m md
F  Eq 
If the particle is initially at rest, then the initial speed u = 0. The final speed, v, is obtained from the
Mechanics formula
v 2  u 2  2as
 02

Vq
d
md
2Vq
m
The final speed is independent of the distance between the plates.
To find out the gain in kinetic energy, we calculate
1 2
mv  qV
2
The energy gained by the charged particle is qV. We could have predicted this – it is the potential
energy difference between the starting point and the end point. Remember that the potential energy is
the potential multiplied by the charge.
The electronvolt
We have seen that the energy gained by a charge q accelerated through a potential V is qV.
At the atomic and sub-atomic level, charge comes in units of e, the electron charge. This is 1.6 × 10-19
C. This can give us an idea of the energy scale involved in accelerating these charges.
If an electron is accelerated through a potential of 1 V, it gains 1.6 × 10-19 J of kinetic energy. This
shows that the joule is not the most suitable unit for calculations involving small charges.
To avoid having factors of 10-19 everywhere, it is convenient to introduce a new unit of energy. The
electronvolt is defined as the energy gained by an electron as it is accelerated through a potential of one
volt.
1 electronvolt = 1eV = 1.6 × 10-19 J
To convert from joules to electronvolts, you need to divide by 1.6 × 10-19. The numbers should get
larger!
Remember that the electronvolt is a unit of energy.
The motion of charges in combined electric and magnetic
fields
It is possible to apply carefully-chosen electric and magnetic fields to a charged particle in such a way
that the resultant force is zero. The fields need to be at right angles to each other and to the direction of
motion of the charge.
The B-field is directed into the page. Note that the directions of the E- and B-fields are different for the
positive and negative charges, but in both cases they may still cancel each other out.
The forces only cancel for particles with a certain velocity, which we can calculate:
FB  FE
Bqv  Eq
v
E
B
The mass (and even the charge) of the particle do not matter! All that counts is the velocity.
This idea can be applied to create a velocity selector. Suppose we have a beam of charged particles
with a range of velocities. If we pass the beam through crossed E- and B-fields, all the particles will be
deflected except those with the correct velocity.
Particles moving too slowly will be deflected more strongly by the E-field; those moving too quickly
will be deflected more strongly by the B-field.
The mass spectrometer
One application where velocity selection is useful is in mass spectrometry. Here, the aim is to
determine the mass of ions. An ion is an atom with too few (or too many) electrons, which means that
it is charged.
The ions are first passed through a velocity selector. Those which emerge from the selector must now
all have the same velocity – these particles are then subjected to a strong magnetic field.
As we have already seen, the path of a charged particle in a magnetic field is a circle of radius
r
mv
Bq
Our particles all have the same velocity, the same charge, and are in the same B-field, so the only thing
which determines the radius of the path is the mass!

The degree of curvature of the path in the B-field therefore allows us to measure the mass.
The cyclotron
Particle physicists have long been interested in observing the effects of smashing particles into each
other at high speeds! If we want to accelerate particles to such high speeds, the simple method outlined
above (linear acceleration) is inefficient.
In a cyclotron, the particles are made to travel in a circle by the application of a magnetic field. They
are then repeatedly “kicked” by the application of an electric field.
As the particle approaches the gap, the electric field across the D-shaped plates is flipped, giving the
electron an energy boost. The magnetic field ensures that the particles continue to travel in a circle.
The electric field must alternate at a particular frequency, which depends on the charge, the mass and
the field strength but not on the speed. In this way, the particles gradually move into wider orbits, and
may be accelerated to quite high energies (MeV, or mega-electronvolts).
8th folder section, first page
Electromagnetic induction
We know that moving charges in a magnetic field experience a force. It is possible for this to create an
electric potential difference or electromotive force (see Electronics notes).
Imagine a straight wire, at right angles to a magnetic field, moving through the field. The direction of
motion is perpendicular both to the wire and to the field. The force on the electrons in the wire will
cause them to move towards one end. The accumulation of charge creates a potential difference
between the ends of the wire.
It is also possible to generate currents this wire, by connecting the ends of the wire to form a loop.
This effect is called electromagnetic induction. It only occurs when changes in flux are taking place.
Magnetic flux
Using our earlier definition of magnetic flux, the flux through this single loop is
  AB
When several loops are joined together to make a coil of N turns, the total flux through the coil
becomes
  NAB
This is because each turn contributes AB to the flux.
The laws of electromagnetic induction
1.
2.
This law is due to Faraday, and concerns the magnitude of the electromotive force (EMF). It
states: “The induced EMF across a conductor is equal to the rate at which magnetic flux is cut
by the conductor.”
Due to Lenz, this law describes the direction of the EMF. It states: “The direction of any
induced EMF is such as to oppose the change which caused it.”
These laws may be combined in the mathematical form
E
E is the induced EMF, while
d
dt
d
is the rate of change of magnetic flux.
dt
Methods of changing the magnetic flux
An EMF is only generated when the magnetic flux changes. There are two ways to change the flux
through a circuit in a magnetic field:
1. Change the field strength while keeping the circuit stationary. The EMF is then
E  A
2.
dB
dt
Keep the field strength constant and move the circuit. The EMF is then
E  B
dA
dt
Self-inductance



When an EMF is applied to a coil of wire, the current through the coil will increase, creating a
magnetic field
This changing magnetic field leads to a change in the magnetic flux through the coil
The changing magnetic flux induces an EMF. Lenz’ law tells us that this EMF must be in the
opposite direction to the original applied EMF
The magnetic field is proportional to the current in the wire:
BI
We can calculate the EMF from the magnetic flux:
  NAB
E
d
dB
  NA
since N and A are constant
dt
dt
Finally, we can relate this to the current in the wire:
E
dI
dt
We define the constant of proportionality between the EMF and the rate of change of the current as the
self-inductance of the coil.
E  L
dI
dt
Self-inductance is given the letter L and is measured in henry (H).
Current growth in an inductance
As we have seen, inductance makes it harder to change the current flowing in a circuit.
The circuit shows the symbol for an inductor. When the power supply is first switched on, the inductor
prevents the current from rising immediately to the final value. Instead, the current tends exponentially
to I0. This is similar to the way the voltage increases across a capacitor.
Eventually, when the current is no longer changing, there is no longer any EMF across the inductor. In
the steady state, the circuit behaves as though the inductor were not there! The inductor only opposes
changes in the current.
We may see this again if we short out the power supply once the steady state has been reached. The
inductor does not allow the current to fall to zero immediately! Instead, the current decays slowly. The
rate of decay is the same as the initial rate of increase:
Inductance in a circuit can be annoying! In many situations, it is desirable to be able to switch the
current on and off very rapidly. This is the case in computers and many other electronic devices.
However, it can also be very useful. Inductors can be used to make resonant circuits and signal filters.