Learn what magnetic fields are and how to calculate them.

What is a magnetic field?

A magnetic field is a picture that we use as a tool to describe how the magnetic force is distributed in the space around and within something magnetic.
When we speak of the force due to a magnet (or any force for that matter) it has to be on something. Strictly speaking a force vector field tells us the magnitude and direction of a force on a small test particle at any point.
With the electric force the small test particle we use is the electron. It turns out that there is no equivalent particle for the magnetic force. The term magnetic monopole is given to such a particle. As far as we know, magnetic monopoles don't exist in nature and all magnetic field sources are dipolar in nature.
Most of us have some familiarity with everyday magnetic objects and recognize that there can be forces between them. We understand that magnets have two poles and that depending on the orientation of two magnets there can be attraction (opposite poles) or repulsion (similar poles). We recognize that there is some region extending around a magnet where this happens. The magnetic field describes this region.
There are two different ways that a magnetic field is typically illustrated:
These descriptions are presented in terms of a 2D slice of the magnetic field and as such could be drawn on a piece of paper. In reality, the magnetic field extends through 3D space, though for gaining a basic understanding of magnetic fields and solving many problems a 2D description is sufficient.
  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.
    A compass is nothing more than a tiny magnet suspended such that it can freely rotate in response to a magnetic field. Like all magnets, a compass needle has a north pole and a south pole that are attracted and repelled by the poles of other magnets. When the compass is placed in a strong magnetic field, the forces of attraction and repulsion turn the needle until it is aligned with the direction of the field.
    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.
      The magnetic field of the Earth arises from moving iron in the Earth's core. The poles of the Earth's magnetic field are not necessarily aligned to the geographic poles. They are currently off by about 10^\circ and over geological periods of time can flip. Currently the magnetic south pole is located near the geographic north pole. This is why the north pole of a compass will point towards it (opposite poles attract).
    • 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?

Because a magnetic field is a vector quantity, there are two aspects we need to measure to describe it; the strength and direction.
The direction is easy to measure. We can use a magnetic compass which lines up with the field. Magnetic compasses have been used for navigation (using the Earth's magnetic field) since the 11ᵗʰ century.
Interestingly, measuring the strength is considerably more difficult. Practical magnetometers only came available in the 19ᵗʰ century. Most of these magnetometers work by exploiting the force an electron feels as it moves through a magnetic field.
Very accurate measurement of small magnetic fields has only been practical since the discovery in 1988 of giant magnetoresistance in specially layered materials. This discovery in fundamental physics was quickly applied to the magnetic hard-disk technology used for storing data in computers. This lead to a thousand-fold increase in data storage capacity in just a few years immediately following the implementation of the technology (0.1 to 100 Gbit/inch2\mathrm{Gbit/inch^2} between 1991 and 2003 [2]). In 2007 Albert Fert and Peter Grünberg were awarded the Nobel Prize in Physics for this discovery.
In the SI system, the magnetic field is measured in tesla (symbol T\mathrm{T}, named after Nikola Tesla). The Tesla is defined in terms of how much force is applied to a moving charge due to the field. A small refrigerator magnet produces a field of around 0.001 T0.001~\mathrm{T} and the Earth's field is about 5105 T5\cdot 10^{-5}~\mathrm{T}. An alternative measurement is also often used, the Gauss (symbol G\mathrm{G}). There is a simple conversion factor, 1 T=104 G1~\mathrm{T} = 10^4~\mathrm{G}. Gauss is often used because 1 Tesla is a very large field.
In equations the magnitude of the magnetic field is given the symbol BB. You may also see a quantity called the magnetic field strength which is given the symbol HH. Both BB and HH have the same units, but HH takes into account the effect of magnetic fields being concentrated by magnetic materials. For simple problems taking place in air you won't need to worry about this distinction.

What is the origin of the magnetic field?

Magnetic fields occur whenever charge is in motion. As more charge is put in more motion, the strength of a magnetic field increases.
Magnetism and magnetic fields are one aspect of the electromagnetic force, one of the four fundamental forces of nature.
There are two basic ways which we can arrange for charge to be in motion and generate a useful 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 rr from a long straight wire carrying current II the equation is
B=μ0I2πrB = \frac{\mu_0 I}{2 \pi r}
Here μ0\mu_0 is a special constant known as the permeability of free space. μ0=4π107 Tm/A\mu_0 = 4\pi\cdot 10^{-7}~\mathrm{T\cdot m / 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.
The right-hand-grip-rule is a useful shortcut, but does have a more fundamental origin as the vector cross product. It is also known as the coffee-mug rule or the corkscrew-rule.
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]
  1. We can exploit the fact that electrons (which are charged) appear
    For understanding the magnetic fields around magnets, it is mostly sufficient to think of an electron like a solid charged ball spinning around a solid nucleus. However, it does lead to a misconception that different electrons could be spinning around at many different speeds and produce many different magnetic fields. It turns out that this is not true; there are only a few possible values of the angular momentum of the electron which are described by the quantum structure of the atom.
    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.
      The physics which describes this pairing was developed by Wolfgang Pauli in 1925 and is known as the Pauli exclusion principle.
    • 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

Figure 5 shows a setup in which a compass is placed near a vertical wire. When no current is flowing in the wire the compass points north as shown due to the Earth's field (assume the field of the Earth is 5105 T5\cdot 10^{-5}~\mathrm{T}).
Figure 5: Compass and wire experiment (viewed from above, no current flowing).
Figure 5: Compass and wire experiment (viewed from above, no current flowing).
Exercise 1a:
What current (magnitude and direction) would be required to cancel out the field of the Earth and 'confuse' the compass?
First we need to arrange Ampere's law for a straight wire to be in terms of current:
B=μ0I2πrB = \frac{\mu_0 I}{2 \pi r}
I=B2πrμ0I = B\frac{2\pi r}{\mu_0}
As shown on the diagram, the distance rr from the compass to the wire is 0.05 m0.05~\mathrm{m}. Substituting in the numbers:
I=(5105 T)(2π)(0.05 m)4π107 Tm/A=12.5 A\begin{aligned}I &= (5\cdot 10^{-5}~\mathrm{T})\frac{(2\pi)\cdot (0.05~\mathrm{m})}{4\pi\cdot 10^{-7}~\mathrm{T\cdot m / A}}\\ &= 12.5~\mathrm{A}\end{aligned}
This is quite a large current. A typical laboratory power supply might only supply up to 3 A3~\mathrm{A}. We also need to find the direction of the field. Using the right-hand-grip-rule we need to point our thumb down in order to have our fingers point in the opposite direction to where the compass is pointing. So the current needs to be flowing into the page in Figure 5.
Exercise 1b:
Suppose our power supply is limited to a total of 1.25 A1.25~\mathrm{A}. Can you suggest an alternative configuration of the experiment which produces the same effect on the compass?
Two options are available to us:
  1. We could simply reduce the distance between the wire and the center of the compass. If we are limited to 1.25 A1.25~\mathrm{A} (one tenth what we had before) then the distance would have to be reduced by the same fraction, i.e. to 5 mm5~\mathrm{mm}. Naturally, this assumes that the body of the compass has radius of 5 mm5~\mathrm{mm} or less.
  2. We could increase the magnetic field by adding more wires, each carrying the same current. Because the current in a long piece of wire is the same everywhere, we can in principle achieve this with a single long piece of wire arranged to allow the current to make 10 'passes'. However, this only works if the current is in the same direction on each 'pass' otherwise the fields will be in opposite directions and cancel each other out. The best way to achieve this is by making a vertical coil with 10 turns and radius large enough that the magnetic field from the opposite side of the coil is small enough to be negligible from the point of view of the compass.

References

[1] Newton Henry Black, Harvey N. Davis (1913) Practical Physics, The MacMillan Co., USA, p. 242, fig. 200 (public domain)
[2] UK Success Stories in Industrial Mathematics. Philip J. Aston, Anthony J. Mulholland, Katherine M.M. Tant. Springer, Feb 4, 2016
[3] This is a file from the Wikimedia Commons. This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International, 3.0 Unported, 2.5 Generic, 2.0 Generic and 1.0 Generic license.
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