In posts 16.16 and 16.25, we met the idea of a field – a region of space in which an object experiences a force that depends on its properties. For example, an object experiences a force in a gravitational field that is proportional to its mass (post 16.16); in an electrical field it experiences a force that is proportional to its charge (post 16.25).
A gravitational field is created by mass; the gravitational field at a distance r from an object of mass M is given by g = GM/r2 where G is the gravitational constant (post 16.16). According to post 16.16, the force acting on an object of mass m at this point was mg.
We now recognise that force is a vector, so that, strictly speaking, mg is the magnitude of the force (post 16.25). How do we represent a field in vector notation?
We can define the gravitational field by the vector equation
g = – GMu/r2
where r is the vector from the centre of mass (post 17.21) of the object of mass M to a point is space (so that r is its modulus) and u is a unit vector in the direction of r. Remember that a unit vector is not a distance – it’s just a vector whose modulus is the number one. So u tells us only about direction and is not measured in the units of distance (like metres).
Now an object of mass m whose centre of mass (post 17.21) is at a position defined by r experiences a force of
F = mg = – GMmu/r2
Why is there a minus sign in the equations above? The vector r points from M to m but the force that acts on m, because it is in the field, is attractive – so it acts in the opposite direction. The minus sign shows that F and r are in opposite directions.
An alternative way to define g would be to write it in the form
g = – GMr/r3.
This is the way it was taught to me when I was a student in London. But I prefer the way I’ve written the definition in my first equation because it clearly shows that Newton’s law of gravitation (post 16.16) is an inverse square law – the field falls of as the square of the distance from M.
The electric field at a point defined from by the vector r from a charge q is
E = qu/(4πεr2)
where ε is the permittivity of the stuff surrounding the point (post 16.25). The force acting on a charge q’ at this point is then
F = qq’u/(4πεr2).
Now there is no minus signs. That’s because if q and q’ are both positive or both negative (“like charges”), the force is repulsive and so acts in the same direction as r. When one of the charges is negative and the other is positive, putting numbers into the equation above gives a negative result – so the force is attractive. So this equation automatically states something about electrostatics that you might have been taught at school – “like charges repel, unlike charges attract”. The vector equation includes this idea without the need for us to remember supplementary rules.
The vector equation for the magnetic field caused by an electric current (post 16.25) also simplifies stuff that you might have learnt at school – but we need to do some more vector algebra first (the subject of a later post). At first, learning about vectors seems to complicate things but, when you get used to the ideas it reduces the need to remember a lot of rules and makes ideas easier to understand.