In this set of notes we will be defining the push or pull produced by a magnetic field (maybe not surprisingly) as the magnetic force. We will continue to look at the most simple case - the force from a magnetic field on a single moving charge, before moving on to consider more complicated situations like a current carrying wire. It turns out that the most complicated case with regards to magnetic forces is actually how one bar magnet attracts another, requiring torques and something we call the magnetic moment. This may be covered in your future courses, but for our purposes we will stick with the magnetic force on a moving charge (or eventually the force on a wire).

In the real world, the magnetic force is actually extremely relevant and has wide-spread applications. For example, the mass spectrometer is largely based on the magnetic force and allows scientists (of all varieties) to distinguish/separate particles that have different masses. In addition, the reason why both the Aurora Borealis and Aurora Australis occur at the poles of the Earth and not in between has to do with the shape of Earth's magnetic field and the force it exerts on charged particles coming from the sun. (This also why you may hear that the Earth's magnetic field acts as a “shield”.) The magnetic force can also be used to create an electric motor or a rail gun or levitate a frog.

The magnetic force is simply a force like any other kind of force (i.e. gravitational force, spring force, or electric force). As such, we already know that the magnetic force:

• has units of newtons (N), where $N = kg\,m/s^2$.
• is a vector - having both a magnitude and direction.
• can be added to other forces on an object to get the net force on the object. Using the net force on the object you can then find the acceleration of the object - you have heard this as the momentum principle or Newton's Second Law of Motion in your Physics 1 class. Mathematically, we represent this relationship as: $$\Delta \vec{p} = \vec{p}_f - \vec{p}_i = \vec{F}_{net,avg} \Delta t$$ $$\vec{F}_{net} = \frac{\Delta \vec{p}}{\Delta t} = \dfrac{d\vec{p}}{dt} = m \frac{d\vec{v}}{dt}= m\vec{a}$$ where $m$ is the mass of the object, $\vec{a}$ is the acceleration of the object, and $\vec{F}_{net}$ is the net force on that object. To find the net force a Free Body Diagram (or a Force Diagram) can be a useful tool.
• acts on an object by something - in the case of the magnetic force this is typically referring to the force from a magnetic field on a charge or current (as we will see in the next pages of notes). NOTE that a moving charge cannot feel a force from the magnetic field it produces (for the same reason why you can't float or fly by pulling on your toes). In order for a moving charge to feel a force, there must be an external magnetic field.

In the next pages of notes, we will go over in detail how to find the magnetic force on a moving charge.