Magnetic fields

Magnetism has been known for much longer than electricity.

  • The Indian surgeon Sushruta described removing iron arrows from patients using a magnet in 600 BCE.1
  • Thales of Miletus, a Greek philosopher, is also said to have been aware of magnets in 600 BCE.
  • The Chinese Qin dynasty was undoubtedly aware of the Earth’s magnetism by 200 BCE and used it for navigation by the 11th century.

Despite this long history and seemingly easier path to discovery, magnetism is harder to understand than electricity.

In common with electric fields:

  • Magnetism is associated with a vector field, $\vec{B}$, with a direction and magnitude at every point in space.
  • The magnetic field $\vec{B}$ is generated by charges.
  • Magnetic fields exert forces on charges.

But, unlike electric fields:

  • There is no magnetic charge; magnetic fields are instead generated by moving electric charges.
  • The force generated by a magnetic field is perpendicular to the field, not along it.

These facts are rather strange and explain why magnetism can be so hard! But it is worthwhile to persevere with learning it because:

  • Magnetism is crucial in modern technology, from electric motors to magnetic resonance imaging.
  • As soon as charges move, the electric field and magnetic field become intertwined, so one cannot understand one without the other.
  • The interplay between electric and magnetic fields is the basis of the theory of light, explains the need for relativity, is the foundation of quantum mechanics. There is no escaping it!

Magnetic fields from a steadily moving charge

Remember from lecture 1 that the electric field generated by a single static charge is given by Coulomb’s law:

\[\vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{\vert \vec{r} \vert^2} \hat{\vec{r}}\,,\]

where $\vec{r}$ is the vector pointing from the charge to the point where we are measuring the field, and $\epsilon_0 \simeq 8.85 \times 10^{-12}\,{\rm C^2\,N^{-1}\,m^{-2}}$ is the permittivity of free space.

Magnetic fields are generated by moving charges. The magnetic field generated by a charge $q$ steadily moving at velocity $\vec{v}$ (provided $\vert \vec{v}\vert $ is not too large) is given by:

\[\vec{B} = \frac{\mu_0}{4\pi} \frac{q}{\vert \vec{r} \vert^2} \vec{v} \times \hat{\vec{r}}\,,\]

where $\mu_0 \simeq 4\pi \times 10^{-7}\,{\rm N\, s^{2}\, C^{-2}}$ is known as the permeability of free space.

  • This is a version of something known as the Biot-Savart law, though we will see a different form in the next lecture. It was discovered experimentally by Jean-Baptiste Biot and Félix Savart in 1820, but first stated in this form by Oliver Heaviside in 1888.
  • Today we understand it as a consequence of the deeper theory of electromagnetism – i.e. it’s not a fundamental law, but it is very useful for calculations.
  • In many ways it is similar to Coulomb’s law, e.g. the field falls off as $1/\vert \vec{r} \vert^2$ and is proportional to the charge $q$.
  • As with the Coulomb law, this expression is not universally valid – but it is valid in a number of situations, including when the charge is moving slowly. That restriction is worth being aware of, but don’t worry about it too much for this course.
  • As with electric fields, magnetic fields superpose, so you can add up the fields from different moving charges, each calculated using the equation, to get the total field.
  • When comparing Coulomb and Biot-Savart laws, the constant of proportionality is different…
  • …but this is not the most important difference. More importantly, the force is also proportional to the velocity with which the charge is moving. (So it is zero for the static case.)
  • …also, the direction of the field is perpendicular to the radial direction, not parallel to it. It is also perpendicular to the velocity of the charge.
  • You can tell this from the expression above because the field is proportional to the cross product of the velocity $\vec{v}$ and the radial direction vector $\hat{\vec{r}}$.

If you don’t recall how cross products work, now is a good moment to revise it.

  • The cross product of two vectors $\vec{a} \times \vec{b}$ is a vector $\vec{c}$ that has magnitude $\vert\vec{a}\vert\vert\vec{b}\vert \sin \theta$ where $\theta$ is the angle between $\vec{a}$ and $\vec{b}$. (Therefore, if $\vec{a}$ and $\vec{b}$ are parallel, the cross product is equal to zero)
  • The direction of $\vec{c}$ is perpendicular to both $\vec{a}$ and $\vec{b}$.
  • For example, if $\vec{i}$, $\vec{j}$ and $\vec{k}$ are unit vectors in the $x$, $y$ and $z$ directions, then $\vec{i} \times \vec{j} = \vec{k}$.
  • Cross products mathematically express the ‘right-hand rule’. If you point your right thumb in the direction of $\vec{a}$ and your right index finger in the direction of $\vec{b}$, then your right middle finger will point in the direction of $\vec{c}$.

When we visualise magnetic fields, they look quite different to electric fields. Here is a magnetic field generated by a charge moving through the plane of your screen. We imagine getting a ‘snapshot’ of the field just as the charge passes through.

  • By convention $\otimes$ symbol indicates a positive charge moving into the page, while a $\odot$ indicates a positive charge moving out of the page.
  • To remember this, think of an arrow-head. If it is coming out of the page you see its tip (a $\odot$), if it is going into the page you see its tail (a $\otimes$).

Magnetic fieldlines have two very different properties from electric fieldlines:

  • They always form closed loops (not just for the simplified case of the single moving charge). This is because there are no magnetic charges, so magnetic fieldlines cannot start or end anywhere.
  • They do not indicate the direction of the force on a charge, in fact the force is always perpendicular to the field as we will see.

Forces on moving charges

The force on a charge $q$ from an electric field $\vec{E}$ is given by $\vec{F} = q\vec{E}$ (lecture 1). The force is parallel to the field.

The force on a charge $q$ from a magnetic field $\vec{B}$ is given by the Lorentz force law:

\[\vec{F} = q\vec{v} \times \vec{B}.\]

where $\vec{v}$ is the velocity of the charge. The force is perpendicular to both the velocity and the magnetic field.

So:

  • Magnetic fields circle around moving charges
  • Moving charges circle around magnetic fields

It’s neat! But also can be quite confusing. The net magnetic force is such that two moving charges are attracted to each other if they are moving in the same direction, and repelled if they are moving in opposite directions. In the lecture we’ll look at working this out algebraically using the properties of the cross product.

Below, the field is shown for the black charge (moving into the page) and the force is shown on the purple charge (also moving into the page). Try moving them around to see how the force changes.

If there is an electric and a magnetic field, the force on a charge is the sum of the two forces:

\[\vec{F} = q(\vec{E} + \vec{v} \times \vec{B}).\]

So actually if you have two like charges moving alongside each other, the electric force still repels them, while the magnetic force tries to attract them.

Discussion point: could the magnetic force ever overcome the electric force of repulsion so that they cancel completely? At what speed would the charges have to move?

Work done by magnetic fields

Because the force is perpendicular to the charge’s velocity, it never does any work on the charge. The rate at which a force does work on a moving object is equal to $\vec{F} \cdot \vec{v}$, which is equal to zero; we’ll see this algebraically in the lecture.

Strength of the magnetic field

The SI units of the magnetic field are the tesla (${\rm T}$). From the Lorentz force law, you can see that this must be equivalent to ${\rm N\,s\,C^{-1}\,m^{-1}}$.

Sometimes (but not in this course) you will see magnetic fields measured in gauss (${\rm G}$), where $1\,{\rm T} = 10^4\,{\rm G}$. That is because magnetic fields are normally much weaker than a tesla. Here are some typical values:

Example Magnetic field strength
Interstellar space $10^{-10}\,{\rm T}$
Earth’s surface $5 \times 10^{-5}\,{\rm T}$
Fridge magnet $0.01\,{\rm T}$
Typical MRI machines $1.5\,{\rm T}$
Strongest magnetic fields produced in a lab $100\,{\rm T}$
Magnetic fields in a typical neutron star $10^{8}\,{\rm T}$
Magnetic fields in a magnetar $10^{11}\,{\rm T}$

Discussion point: typical electric fields in everyday situations are small because matter is nearly neutral (see lecture 2). But why are strong magnetic fields rare?

Footnotes

  1. He followed up by saying that, by contrast, “a shaft of grief, driven into the heart” should be “removed by exhilaration and merry-making”. ↩