Section 20.1 in Matter and Interactions (4th edition)

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[[184_notes:q_b_force|Previous Page: Magnetic Force on Moving Charges]]*/

===== Path of a Charge through a Magnetic Field =====
We just talked about the force that a moving charge feels when it travels through a magnetic field. So now the question remains: what happens to the charge when it feels this force? Since the magnetic force is perpendicular to the velocity of the charge, **we will show that the charge actually begins to move in a circular pattern.**

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===== Path of the Moving Charge =====

[{{  184_notes:week11_3.png?150|Force felt by a charge moving through a B field in the -x direction}}]

Let's start by considering a charge moving to the left ($-\hat{x}$ direction) in a magnetic field that points into the page ($-\hat{z}$ direction). Using the right hand rule, we can figure out that the force on this charge is down ($-\hat{y}$ direction). So what does this mean for the path of our charge? We can use the [[183_notes:momentum_principle|momentum principle]] to help us figure it out. The momentum principle says that:
$$\Delta \vec{p} = \vec{p}_f - \vec{p}_i = \vec{F}_{net,avg} \Delta t$$
where momentum is simply $\vec{p}=m\vec{v}$ (meaning momentum points in the same direction as velocity). We can rewrite the momentum principle into it's "update form" by saying:
$$\vec{p}_f  = \vec{F}_{net,avg} \Delta t + \vec{p}_i$$

[{{184_notes:week11_4.png?150|Force changes the direction of the velocity  }}]
This means that if our magnetic force is pushing the charge down for a short time, the velocity of the charge will then point slightly down. If we do the same process at the next location and the next location (it turns out a computer is pretty good at doing these calculations), we find that **the charge will move in a circle**, with the magnetic force always pointing toward the center of the circular path. It turns out that this is a perfect example of [[183_notes:ucm|uniform circular motion]]. 


===== Work Done =====
[{{  184_notes:week11_5.png?400|Trajectory of a particle moving through a constant magnetic field at a constant speed}}]

You may be wondering if there is a magnetic potential energy associated with the magnetic field (after all [[184_notes:pc_energy|we did use the electric force to get to electric potential energy]]). We can figure this out by first looking at the work done by the magnetic field on the charge. Remember that [[183_notes:work_by_nc_forces|the general equation for work]] is given by:
$$W_{total} = \int_i^f \vec{F}\cdot d\vec{r}$$
In this case, the force is the magnetic force, which we have already said points perpendicular to the path ($d\vec{r}$) at every point in the circle.
$$W_{B}=\int_i^f \vec{F}_{B}\cdot d\vec{r}$$
Since the dot product is between the magnetic force and the path direction, and because those vectors are perpendicular, this tells us that **the work done by the magnetic field is actually zero**.
$$W_{B}=0$$
This is an important result because this tells us that **there is no change in the energy of the particle.** In other words, the magnetic force **only changes the direction of the moving charge - it does not slow down or speed up as it travels through the magnetic field** (unless there is a non-magnetic force also acting on it). The magnetic field can only turn a charge not accelerate it.

This also means that we do not have a way to define a magnetic potential or a magnetic potential energy in the same way that we did with electric fields. It turns out that we can define a [[https://en.wikipedia.org/wiki/Vector_potential|vector potential]], which you may talk about in your future courses.

===== Examples =====
  * [[:184_notes:examples:Week10_radius_motion_B_field|Radius of Circular Motion in a Magnetic Field]]
  * [[:184_notes:examples:Week10_helix|Helical Motion in a Magnetic Field]]
    * Video Example: Helical Motion in a Magnetic Field
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