# Scattering with magnetic monopole

## Group 16
* 劉芷辰 B09202053
* 李逸寬 B10202045
* 王彥傑 B10202042
* 陳為霖 B10202044

## Abstract
In Classical Mechanics, we have learned to analyze the system with central force[^Marion]. In this project, we want to study a motion with a different kind of force, whose direction is not along the line connecting the centers of the objects. To be more specific, we want to see how an electric charge move when there is a magnetic monopole. To simplify the question, the position of the monopole is fixed, and we only consider the motion of the electric charge. The process is not too difficult since we have already gotten the law of motion. We can derive some important results. For example, the conserved quantity, and the physical quantities associated with scattering. It can be shown that the original angular momentum is no longer conserved. Besides, the scattering process has unintuitive phenomenon. To visualize the motion and compare the result given by the computer with our calculation, we use VPython to simulate the motion. Moreover, it can help us to imagine the system with 3D animations.
## Background and Objectives

From electromagnetism[^Griffiths], we learn that we have not found magnetic monopoles in reality. However, it can be interesting to investigate the dynamics with monopoles[^Shnir]. The field created by a magnetic monopole $g\frac{\mathbf{r}}{r^3}$ is similar to electric field created by an electric charge $e\frac{\mathbf{r}}{r^3}$, but the law of force with magnetic field is different from the one with electric charge. We have $e\mathbf{v}\times\mathbf{B}$ instead of $e\mathbf{E}$. From the magnetic force, it can be shown that we have to modify the angular momentum of the system, so that it can become a conserved quantity. To elaborate, we first have the original angular momentum
$$\mathbf{L}_0=m\mathbf{r}\times\mathbf{v},$$
and we also have the law of motion
$$m\frac{d^2\mathbf{r}}{dt^2}=e\mathbf{v}\times\mathbf{B}=\frac{eg}{r^3}(\frac{d\mathbf{r}}{dt}\times\mathbf{r}).$$
Then we have
$$\mathbf{r}\cdot\frac{d^2\mathbf{r}}{dt^2}=\frac{1}{2}\frac{d^2(r^2)}{dt^2}-v^2,$$
so that one can write
$$r=\sqrt{v^2t^2+b^2},$$
where $b$ is the minimal distance from the electric charge to the monopole, the impact parameter.
What's more, with the law of motion, we can derive that
$$\frac{d\mathbf{L}_0}{dt}=m\mathbf{r}\times\frac{d^2\mathbf{r}}{dt^2}=eg(\frac{d\mathbf{r}}{dt}\frac{1}{r}-\frac{\mathbf{r}}{r}(\hat{\mathbf{r}}\cdot\frac{d\mathbf{r}}{dt})=\frac{d}{dt}(eg\hat{\mathbf{r}}).$$

Thus, we have the modified angular momentum $\mathbf{L}=\mathbf{L_0}-eg\hat{\mathbf{r}}$, and it will not change with time. It is totally different from the system which only has an electric field. We can also show that $\mathbf{L}\cdot\hat{\mathbf{r}}=-eg=const.$. Interestingly, the charge will move on the surface of a cone, and the axis of the cone is directed along $-\mathbf{L}$. Let the direction of $-\mathbf{L}$ be $\hat{z}$, then we can have the geometry of scattering (Fig. 1). We have defined $\cot\theta=\frac{eg}{mvb}$. Unlike the scattering in a Coulomb field,

In this project, we are going to discuss more about the scattering of an electric charge in the presence of a magnetic charge. The final objective is to simulate the scattering with VPython.

<figcaption>Fig. 1: The geometry of scattering. </figcaption>

## Methods, Steps and Progress

### Methods
We will use VPython to simulate the motion of an electron under the magnetic field of a monopole.

### Timetable

|Week|Task |
|----|------|
|1 |Reading papers and discussion. |
|2|Reading papers and discussion.|
|3|learning VPython.|
|4|learning VPython.|
|5|coding the simulation with VPython.|
|6|further discussion.|
|7|Designing poster layout.|
|8|practicing for the oral presentation.

### Responsibilities
* discussion: 王彥傑、劉芷辰
* simulation program: 李逸寬、陳為霖
* poster: 王彥傑、劉芷辰、李逸寬、陳為霖
* oral presentation: 王彥傑、劉芷辰、李逸寬、陳為霖

## Expected Difficulties and Solutions
When studying Electrodynamics, we learnt that the Lagrangian of a charged particle under electromagnetic field can be written as
$$\mathcal{L}=\dfrac{1}{2}m\dot{\mathbf{r}}^2+e\dot{\mathbf{r}}\cdot\mathbf{A}-e\phi$$
where
$$\partial_t\mathbf{A}-\nabla\phi=\mathbf{E},\qquad \nabla\times\mathbf{A}=\mathbf{B}.$$
However, it is impossible to solve the equation
$$\nabla\times\mathbf{A}=g\frac{\mathbf{r}}{r^{3}}$$
in the whole region with a single potential $\mathbf{A}$ since the divergence of the LHS vanishes everywhere while the divergence of the RHS does not.

With some effort, we can write down a possible form of $\mathbf{A}$ in the region $\mathbb{R}^3-\{(0,0,a)|a\in\mathbb{R}^+\}$ as
$$\mathbf{A}=(g\dfrac{1+cos\theta}{rsin\theta}sin\varphi,-g\dfrac{1+cos\theta}{rsin\theta}cos\varphi,0)=\dfrac{g}{r}\dfrac{\mathbf{r}\times\hat{\mathbf{z}}}{r-\mathbf{r}\cdot\hat{\mathbf{z}}}.$$
We can see that this potential satisfies our requirement perfectly without a string starting from the origin and ending at $(0,0,\infty)$. This strange property implies that there are something interesting behind that and we need to discuss the potential around the string more carefully. This string is called "Dirac string", and things become intriguing after considering the gauge of the potential.

## Results and Evaluation
We have written down the needed elements in our background. Therefore, it is not difficult to use them to discuss more about the motion. As a start, we can try to write down the explicit form of the velocity
$$\mathbf{v}=\frac{d\mathbf{r}}{dt}=\frac{1}{mr^2}(\mathbf{L}\times\mathbf{r})+\frac{v}{\sqrt{1+(b/vt)^2}}\hat{r},$$
then we can have the angular velocity
$$\omega=\frac{L}{mr^2}=\frac{\sqrt{(mvb)^2+(eg)^2}}{m(v^2t^2+b^2)}=\frac{d\varphi}{dt},$$
where $\varphi$ is the azimuthal angle. By integration, we can obtain
$$\varphi=\frac{1}{\sin\theta}\arctan\frac{vt}{b}.$$
Asymptotically, we have the direction of velocity at $t=\pm\infty$ (Fig. 2[^Shnir])
$$\hat{\mathbf{v}}\mid_{t=\pm\infty}=(\pm\sin\theta\cos\frac{\Delta\varphi}{2}, \sin\theta\sin\frac{\Delta\varphi}{2},\pm\cos\theta),$$
where $\Delta\varphi=\varphi(\infty)-\varphi(-\infty)=\pi/\sin\theta$. Consequently, we can get an important result, the scattering angle $\Theta$ with a magnetic monopole
$$\cos\Theta=\hat{\mathbf{v}}\mid_{t=\infty}\cdot\hat{\mathbf{v}}\mid_{t=-\infty}=2\sin^2\theta\sin^2(\frac{\pi}{2\sin\theta})-1$$
$$\Rightarrow \Theta=2\arccos(\sin\theta\mid\sin(\frac{\pi}{2\sin\theta})\mid),$$
where $\sin\theta=mvb/\sqrt{(mvb)^2+(eg)^2}$. Obviously, the scattering angle is not a monotonous function of the impact parameter.
To produce the simulation of scattering, we will use VPython to show the motion and check whether the scattering angle of the particle matches the theoretical results or not. One of the goal is to see if we can simulate the non-monotonic scattering angle on the computer. In addition, we will try to find closed orbitals under the magnetic field of the magnetic monopole. However, it might be hard to find such an orbital because of the round-off errors in the simulation program.
<figcaption>Fig. 2: The direction of velocity. </figcaption>

## References
[^Shnir]:Ya. Shnir. (2005). Magnetic Monopoles. Springer.
[^Marion]:Stephen T. Thornton, Jerry B. Marion. (2004). J. Classical Dynamics of Particles and Systems. Brooks/Cole-Thomson Learning.
[^Griffiths]:David J. Griffiths. (2013). Introduction to Electrodynamics. Pearson.