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FIELDS
These notes are an introduction to fields in physics.
FIELDS OF FORCE
If you put two magnets near each other, you can feel a force between them. This is strange, in that they affect
each other even though they are not touching each other. This works in a vacuum, so it’s not the air in
between which carries the effect. We explain this – in a limited way – by saying magnets have a ‘force field’
around them.
The effect of a bar magnet seems to be concentrated at the ends, and we say the magnet has a pole at each
end. A magnet is affected by the Earth’s magnetic field and lines up so that one end points north and the other
south. A bar magnet has a north pole at one end, and the other has a south pole.
Two north poles repel each other, and two north poles. A north and a south attract each other.
VISUALIZING FIELDS
If tiny fragments of iron are scattered around a magnet, they arrange themselves in a pattern, which looks like
this:
There appear to be lines going from one pole to the other. If you had a small north pole by itself, it would
move along these lines, away from the magnet north pole and to the south. These are called ‘lines of force’ or
‘field lines’.
Usually the pattern is made by scattering iron filings on a piece of paper on a magnet. But in fact the field is
three dimensional. There is also the question of what the field is like inside the magnet.
GRAVITATIONAL FIELDS
A mass also produces a field around it, just like a north pole. Two masses
attract each other, just like a north and a south pole. However the force is
normally weak, so we do not notice it. Only in the case of very large
masses, like the Earth, Sun and Moon, is it large enough to notice. We call
the force of gravity on a mass due to the Earth ‘weight’. It was Isaac
Newton who realized this was a universal force between masses, and he
could use it to explain the movement of the Moon in orbit around the
Earth.
The lines of force produced by a mass are pretty dull. If we imagine
another small ‘test’ mass near it, there will be a force on the test mass towards the big mass, and the test
mass would move simply straight to the big central mass. So the field lines are straight lines spreading out
from the centre.
So this is the gravitational field around, for example, the Sun. What about inside the Sun. To start with, it is
simpler just to think about the Sun as a point mass – all the mass concentrated at a single point in space. In
reality the Sun is pretty big – but we pretend it is a point mass to start with.
FIELD STRENGTH
As well as the pattern of field lines, we can also think about how strong the field is. This is the field strength or
intensity. The gravitational force on a test mass depends on how big the test mass is. So we define the field
strength as the force on a unit mass – one kilogram.
At the surface of the Earth, the field strength is about 9.81 Newtons per kilogram. In other words, each
kilogram of mass has a gravitational force on it of 9.81 Newtons.
Newton proposed a way of calculating gravitational field strength – as
where M is the central mass, r is the distance you are away from it, and G is the ‘constant of gravitation’.
EXAMPLE
Let’s calculate the gravitational field strength of the Sun at the Earth’s orbit. In SI units, the value of G is 6.67 X
-11
2
-2
30
10 (the units are N m kg ). The mass of the Sun is 2 X 10 kg. The distance from the Earth to the Sun is
11
about 1.5 X 10 m. So the field strength is
The field due to the Earth at its surface is 9.81 N/kg. So the Sun’s field intensity is less than one thousandth of
that – because the Sun is further away. But it does have a noticeable effect, in the form of tides.
INVERSE SQUARE LAW FIELDS
2
Gravitational field strength varies as 1/r – it is an inverse square law field. If you are twice as far away, gravity
is one quarter the strength.
Why do we think this is true? One reason is that experiments measure it like this. Secondly, an inverse square
law fits with the shape of the orbits of the planets. These move in ellipses, which the maths predicts you get
for inverse square law fields.
But why is it true? One answer is because of the way space works. Suppose you have one sphere, and double it
in size. Then the surface area increases four-fold. If we think of gravity spread over the sphere, it will be four
times as spread out over a sphere of twice the size. Hence the inverse square law.
DIPOLES
For magnets we have north and south poles – two kinds of poles. But for gravity we just have one kind of
mass. Magnetism and gravity are different.
There are no magnetic monopoles. In other words, no-one has ever found a north pole without a south pole.
Magnetic poles occur in pairs, or dipoles.
While for gravity, there is only positive mass. Corresponding to north and south poles, we might have positive
and negative mass – but we do not. Negative mass would be weird – it would accelerate in the opposite
direction to a force upon it. Push it left, and it goes right. Only there is no negative mass.
So there are no gravitational dipoles. But the idea is still used. Suppose we have a planet inside which the mass
is not perfectly uniformly distributed – which in practice is always the case, not least because of mountains
and valleys. Suppose we have a planet where the northern hemisphere is slightly more massive than the
southern. We can model this as a point mass (most of the planet mass) together with a small dipole – which
adds some mass in the north, and subtracts some from the south.
DIPOLE FIELD PATTERNS
A bar magnet is a magnetic dipole – a pair of opposite poles. So a dipole field should be like that from a bar
magnet. Here it is:
Imagine this is a bar magnet with a north pole at the top and
south pole below. Where would a small test north pole be
pushed?
Suppose it starts somewhere above the north pole. It will be
repelled, upwards, away from the north. And it will be
attracted, downwards, by the south pole beneath. But the
north pole is closer, so the repulsion is stronger than the
attraction. So the field lines go upwards.
Suppose it is at the equator somewhere. It will be repelled, out
and down, by the north. It will be attracted, in and down, by the
south. If we combine the two forces, we will get a net
downwards force. So the field on the equator goes down.
Below the south pole, the situation is the reverse of above the north – we get a large attraction from the close
south pole, and a weaker repulsion from the further north pole. So the net force would be up.
The shape of a gravitational dipole field is identical.
FIELDS FROM ELECTRIC CHARGES
This is about electrostatic charge – electric charge which does not
move (unless we get a big spark as from a Wimshurst machine). A
moving electric charge is an electric current, and this complicates
things because an electric current produces a magnetic field. So we are
just thinking about electric fields produced by stationary electric
charges.
We can have single electric charges, positive or negative. So we can
have electric monopoles. And we can have pairs, so electric dipoles
exist.
Electric fields are inverse square law, just as for gravity. So the shape of electric fields is identical to that of
gravitational fields.
POTENTIAL
Imagine the gravitational field due to a large point mass such as the Sun. And think about a small test mass.
Suppose we start off with the test mass a long way away, and at rest. Think about its energy. It will have
different amounts of kinetic and potential energy.
Suppose we say it starts at a large distance with zero energy. It is attracted (weakly) to the central mass.
Suppose we let it move in, very slowly, towards the centre. We can arrange for the test mass to do work as it
moves in (work is mechanical energy: work = force X distance). It moves very slowly, so it has zero kinetic
energy. It gives us energy as work, so it loses energy (because energy is conserved). It started with zero – so
now it has less than zero – it has a negative amount of energy.
As it moves in, the gravitational field gets stronger. It will do more work, so its potential energy is increasingly
negative.
At each point, it will have a certain amount of potential energy. How much it has will depend on its mass. If it
is a unit mass, the potential energy at a point is called the potential there.
The potential at a point is the potential energy a unit mass would have, if it had zero initially, a long way away
(‘at infinity’).
Field strength is the force on a unit mass (or charge or pole) at
that point. Forces have magnitude and direction, so they are
vectors. Force fields are therefore vector fields, with a
magnitude and direction at each point. Potential is not a vector.
It only has magnitude.
A visualization of this is shown left. This shows the potential
around a point mass, over the surface of a plane. The colour
goes more red as we get nearer the mass, as the potential
becomes increasingly negative.
The second image also shows the potential, over the surface of
that plane. The ‘vertical’ value indicates the potential at each
point. We can see the gravitational potential well surrounding
the mass.
The gravitational field strength is inverse square:
If you do the maths you find the potential is
2
For points, potential varies as 1/r rather than 1/r .
In the above plot, the plane does not exactly go through the central point mass, so we do not get to r=0. If we
had, the potential would get to -∞. But actual point masses do not exist – don’t mention black holes.
Field intensity is the force on a unit mass, so it is a vector – it has magnitude and direction. Potential is the
potential energy of a unit mass, so it is a scalar – magnitude but no direction. Being a scalar, potential is
mathematically a bit simpler than field strength, and so is widely used.
DIPOLES AND QUADRUPOLES
A dipole is a pair of opposite electrical charges, or magnetic poles, or normal and negative mass.
A quadrupole is a pair of
dipoles. These can be
arranged in various ways,
including as shown here, a
linear quadrupole where
the two dipoles are in line.
The field of a quadrupole is
simply the net effect of the
four charges. This is shown
on the left, and the
potential on the right.