What is a magnetic field?

1. The magnetic field is described mathematically as a vector field. This vector field can be plotted directly as a set of many vectors drawn on a grid. Each vector points in the direction that a compass would point and has length dependent on the strength of the magnetic force.

   [Explain compasses]

   Arranging many small compasses in a grid pattern and placing the grid in a magnetic field illustrates this technique. The only difference here is that a compass doesn't indicate the strength of a field.

   Figure 1: Vector field plot for a bar magnet

   Figure 1: Vector field plot for a bar magnet.
2. An alternative way to represent the information contained within a vector field is with the use of field lines. Here we dispense with the grid pattern and connect the vectors with smooth lines. We can draw as many lines as we want.

   Figure 2: Field line plot for a bar magnet

   Figure 2: Field line plot for a bar magnet

   The field-line description has some useful properties:
   • Magnetic field lines never cross.
   • Magnetic field lines naturally bunch together in regions where the magnetic field is the strongest. This means that the density of field lines indicates the strength of the field.
   • Magnetic field lines don't start or stop anywhere, they always make closed loops and will continue inside a magnetic material (though sometimes they are not drawn this way).
   • We require a way to indicate the direction of the field. This is usually done by drawing arrowheads along the lines. Sometimes arrowheads are not drawn and the direction must be indicated in some other way. For historical reasons the convention is to label one region 'north' and another 'south' and draw field lines only from these 'poles'. The field is assumed to follow the lines from north to south. 'N' and 'S' labels are usually placed on the ends of a magnetic field source, although strictly this is arbitrary and there is nothing special about these locations.

     [Explain magnetic field of the Earth]
   • Field lines can be visualized quite easily in the real world. This is commonly done with iron filings dropped on a surface near something magnetic. Each filing behaves like a tiny magnet with a north and south pole. The filings naturally separate from each other because similar poles repel each other. The result is a pattern that resembles field lines. While the general pattern will always be the same, the exact position and density of lines of filings depends on how the filings happened to fall, their size and magnetic properties.

     Figure 3: Magnetic field lines around a bar magnet visualized using iron filings.

     Figure 3: Magnetic field lines around a bar magnet visualized using iron filings.

How do we measure magnetic fields?

What is the origin of the magnetic field?

1. We make a current flow through a wire, for example by connecting it to a battery. As we increase the current (amount of charge in motion) the field increases proportionally. As we move further away from the wire, the field we see drops off proportionally with the distance. This is described by Ampere's law. Simplified to tell us the magnetic field at a distance $r$r from a long straight wire carrying current $I$I the equation is

Here $\mu_0$mu, start subscript, 0, end subscript is a special constant known as the permeability of free space. $\mu_0 = 4\pi\cdot 10^{-7}~\mathrm{T\cdot m / A}$mu, start subscript, 0, end subscript, equals, 4, pi, dot, 10, start superscript, minus, 7, end superscript, space, T, dot, m, slash, A. Some materials have the ability to concentrate magnetic fields, this is described by those materials having higher permeability.

Since the magnetic field is a vector, we also need to know the direction. For conventional current flowing through a straight wire this can be found by the right-hand-grip-rule. To use this rule imagine gripping your right hand around the wire with your thumb pointing in the direction of the current. The fingers show the direction of the magnetic field which wraps around the wire.

[Explain]

Right-hand-grip rule used to find the direction of the magnetic field (B) based on the direction of a current (I). [3]

Figure 4: Right-hand-grip rule used to find the direction of the magnetic field (B) based on the direction of a current (I). [3]

2. We can exploit the fact that electrons (which are charged) appear
   [explain appear]
   to have some motion around the nuclei of atoms. This is how permanent magnets work. As we know from experience, only some 'special' materials can be made into magnets and some magnets are much stronger than others. So some specific conditions must be required:

• Although atoms often have many electrons, they mostly 'pair up' in such a way that the overall magnetic field of a pair cancels out. Two electrons paired in this way are said to have opposite spin. So if we want something to be magnetic we need atoms that have one or more unpaired electrons with the same spin. Iron for example is a 'special' material that has four such electrons and therefore is good for making magnets out of.

  [Explain 'pairing up']
  • Even a tiny piece of material contains billions of atoms. If they are all randomly orientated the overall field will cancel out, regardless of how many unpaired electrons the material has. The material has to be stable enough at room temperature to allow an overall preferred orientation to be established. If established permanently then we have a permanent magnet, also known as a ferromagnet.
  • Some materials can only become sufficiently well ordered to be magnetic when in the presence of an external magnetic field. The external field serves to line all the electron spins up, but this alignment disappears once the external field is removed. These kinds of materials are known as paramagnetic.
    The metal of a refrigerator door is an example of a paramagnet. The refrigerator door itself is not magnetic, but behaves like a magnet when a refrigerator magnet is placed on it. Both then attract each other strongly enough to easily keep in place a shopping list, sandwiched between the two.

Canceling the field of the Earth

What current (magnitude and direction) would be required to cancel out the field of the Earth and 'confuse' the compass?

[Solution]

Suppose our power supply is limited to a total of $1.25~\mathrm{A}$1, point, 25, space, A. Can you suggest an alternative configuration of the experiment which produces the same effect on the compass?

[Solution]

References