Test

December 1st, 2007

Question: Write down the equation of motion for a charged particle in uniform, parallel electric and magnetic fields, both in the z-direction, and solve it, given that the particle starts from the origin with the velocity (v_0,0,0) . A screen is placed at x=a where $a \ll \frac{mv}{qB}$ . Show that the locus of points of arrival of particles with given m and q, but different speeds v_0 , is approximately a parabola.

Answer: The equation of motion is given by the Lorentz Force equation:

$m\ddot{r} = q \vec{E} + q \dot{r} \times \vec{B}$

However, since $\vec{E} = E \hat{k}$ and $\vec{B} = B \hat{k}$ for this problem, the equation of motion becomes

$m\ddot{r} = q E \hat{k} + q \dot{r} \times B \hat{k}$

which reduces to

$m \ddot{x} = q\dot{y}B, m\ddot{y} = - q\dot{x}B, m \ddot{z} = qE$

First, lets solve $\ddot{z} = \frac{qE}{m}$ . If we integrate $\ddot{z}$ , we get

$\dot{z}(t) = \int \frac{qE}{m}dt = \frac{qE}{m}t + C$

since the initial velocity at t=0 is (v_0,0,0)

, this means $\dot{z}(0) = 0$ .

Therefore, $\dot{z} = \frac{qE}{m}t$ .
If we integrate again, we get:

$z(t) = \int \frac{qE}{m}tdt = \frac{qE}{2m}t^{2} +C$

since the initial position at t=0 is (0,0,0)

, this means z(0) = 0

Therefore, $z = \frac{qE}{2m}t^{2}$ .

Next, we will reduce the other two equations

$\ddot{x} = \frac{qB}{m}\dot{y}$

$\dot{x} = \int \frac{qB}{m}\dot{y}dt = \frac{qB}{m}y + C$

since $\dot{x}(0) = v_0$ , therefore $\dot{x} = \frac{qB}{m}y + v_0$ .

We can similar reduce $\ddot{y}$

$\ddot{y} = -\frac{qB}{m}\dot{x}$

$\dot{y} = -\int\frac{qB}{m}\dot{x}dt = -\frac{qB}{m}x +c$

since $\dot{y}(0) = 0$ , therefore $\dot{y} = -\frac{qB}{m}x$ .

If we put these equations into the original equations of motion, we can decouple the x and y terms.

$\ddot{x} = \frac{qB}{m}(-\frac{qB}{m}x) = - (\frac{qB}{m})^{2}x$

which has a general solution of

$x = Asin(\frac{qB}{m}t) + Bcos(\frac{qB}{m}t)$

However, since x(0) = 0 , therefore $x = Asin(\frac{qB}{m}t)$ .
In order to calculate A, we need to take the derivative.

$\dot{x} = \frac{AqB}{m}cos(\frac{qB}{m}t)$

since $\dot{x}(0) =v_0$ , the following reduction can be made $\frac{AqB}{m} = v_0 \Rightarrow A = \frac{m v_0}{qB}$ .

Therefore, $x = \frac{m v_0}{qB}sin(\frac{qB}{m}t)$ and $\dot{x} =v_0 cos(\frac{qB}{m}t)$ .

Also, since $\dot{x} = \frac{qB}{m}y +v_0$ , we can solve for y

$v_0 cos(\frac{qB}{m}t) = \frac{qB}{m}y + v_0$

$y = \frac{v_0 m}{qB}(cos(\frac{qB}{m}t) - 1)$

This means the equation of motion is:

$x = \frac{m v_0}{qB}sin(\frac{qB}{m}t)$

$y = \frac{v_0 m}{qB}(cos(\frac{qB}{m}t) - 1)$

$z = \frac{qE}{2m}t^{2}$

Comments(1) - Uncategorized

Beowulf coming to Theaters

October 14th, 2007

Every once in a while, I go to Apple’s website to check out the newest movie trailers. I find it is a really good way to check out what movies I want to see. This weekend I came across a trailer for Beowulf which will be coming to theaters on November 16th. For those who haven’t heard of Beowulf [wp], it is an influential Old English heroic epic poem which dates back to 700–750 AD. Beowulf was basis for J.R.R Tolkien’s academic career which was the first serious literature critiques of the poem. The poem was, also, a strong influence in The Lord of the Rings which lead to the development of modern fantasy. Although I would never read Beowulf, I think it is an excellent premise for a movie.

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Finally! A good Text Editor for Windows.

October 6th, 2007

About a couple months ago, I switched text editors from Textpad to E – TextEditor. Despite E – TextEditor’s lame name, I was shocked to find out how superior it was to my previous editor. Although E has some stability issues, I still think that it is one of the best text editors for Windows. However, considering that E was released only 3 months ago, I think it is remarkably stable. Also, because of E’s rapid development and its large community, I would expect that these issues will be quickly resolved. If you are looking for a new text editor for Windows, I would strongly encourage you to try the 30-day trial.

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VanLan 2007

September 24th, 2007

Last weekend I went to VanLan 2007! If you haven’t heard of it, I am sure you are not alone. Every year all my old High School friends go to Vancouver to play computer games and hang-out with each other. It is a good way to stay connected with each other. This year there was only about a dozen of us (a lot less from last year), but it was still a lot of fun. The only down-fall of VanLan is that every year there seems to be more drinking and less computer games. I guess it would be less of an issue if I was a bigger drinker. However, on the positive side, it is a lot easier to beat an intoxicated opponent. Even with this set-back, we did play a lot of computer games. Here is an overview of some of the games we played: Dawn of War, WarCraft 3 mods, Aliens vs Predator 2, Guitar Hero Encore: Rocks the 80s, and much much more.

I was, also, introduced to a cool little game called Geometry Wars. The best way to describe the game is Asteroids on Ecstasy. I never thought that such a simple game could be so visually stimulating… or that hard. The best part is the full game only cost four bucks! Whenever I had some free time on the weekend I was playing this game, so I got my money’s worth already.

Another interesting game I was introduced to was the Starcraft boardgame. I really don’t like Starcraft, so I was shocked to find out how good this game was. Starcraft (the computer game) has absolutely no strategy in it. It is all about micro-management and cruiser rushes. I thought the boardgame would be doomed to fail. However, I was surprised how much strategy was in the game. The best way to describe the game is a cross between Axis vs Allies and Settlers of Catan which are both excellent games. I am definitely tempted to buy the game.

Comments(0) - Computer Games, Informational

Addicted to Hereos

September 9th, 2007

As you can tell, I haven’t had much time to blog lately. I usually try to blog something every week, but that doesn’t always turn out. During the last week, I have been watching the Hereos TV series. My brother lent me the first season last weekend and I already have watched it. It was absolutely amazing. I couldn’t stop watching it and hence why it only took me 4 days to complete the series. It would have taken me a lot less time if I didn’t have to work. If you haven’t seen it yet, you really should watch it. It is in the same league as 24 and Battlestar Galactica. I love it when a TV series has a continuous plot line instead of having a different plot every episode. Your plot line can only go so deep when it has to be contained in 30 minutes (or an hour).

Anyways, the second season starts on September 24. I really hope NBC doesn’t drop the ball on this series. It has so much potential! The last thing we want is another TV series as bad as Lost.

Comments(2) - TV

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Beowulf coming to Theaters

Finally! A good Text Editor for Windows.

VanLan 2007

Addicted to Hereos