Thornton and Marion, Chapter 2, Prob. 22

  • Thread starter union68
  • Start date
  • #1
140
0

Homework Statement



This is a four part problem -- the only issue I have is on part (c) so I'll condense the question:

If the equation of motion of a particle of charge [tex]q[/tex] in an electric field and magnetic field is

[tex]qv_y B \hat{\mathbf{i}} + \left( qE_y - qv_xB \right) \hat{\mathbf{j}} + qE_z \hat{\mathbf{k}} = m \frac{d}{dt} \left( v_x \hat{\mathbf{i}} +v_y \hat{\mathbf{j}} + v_z \hat{\mathbf{k}}\right), [/tex]

obtain expressions for [tex] v_x \left(t\right)[/tex] and [tex] v_y \left(t\right)[/tex]. Show that the time averages of these velocity components are

[tex]\left\langle v_x\left(t\right)\right\rangle = \frac{E_y}{B}[/tex]

and

[tex] \left\langle v_y\left(t\right)\right\rangle = 0[/tex]

(Show that the motion is periodic and then average over one complete period.)

Homework Equations





The Attempt at a Solution



Solutions to the DE are

[tex]v_x \left(t\right) = \frac{E_y}{B} + C_1 \cos \left(\omega t\right) + C_2 \sin \left(\omega t\right)[/tex]

and

[tex] v_y \left(t\right) = -C_1 \sin \left(\omega t\right) + C_2 \cos \left(\omega t\right)[/tex],

where [tex]\omega[/tex] is the cyclotron frequency. Now, clearly the velocity functions are periodic with period [tex]2\pi/\omega[/tex]. What do they mean "time averages," and what does the [tex]\left\langle \right\rangle[/tex] notation mean?

Perhaps,

[tex] \left\langle v_x\left(t\right)\right\rangle = \frac{ v_x\left(0\right) + v_x \left(2\pi/\omega\right)}{2}[/tex],

like an arithmetic mean, but I'm just guessing based on the hint (and that doesn't yield the correct answer). Do they mean average velocity?
 

Answers and Replies

  • #2
tiny-tim
Science Advisor
Homework Helper
25,836
251
Hi union68! :smile:
Solutions to the DE are

[tex]v_x \left(t\right) = \frac{E_y}{B} + C_1 \cos \left(\omega t\right) + C_2 \sin \left(\omega t\right)[/tex]

and

[tex] v_y \left(t\right) = -C_1 \sin \left(\omega t\right) + C_2 \cos \left(\omega t\right)[/tex],

where [tex]\omega[/tex] is the cyclotron frequency. Now, clearly the velocity functions are periodic with period [tex]2\pi/\omega[/tex]. What do they mean "time averages," and what does the [tex]\left\langle \right\rangle[/tex] notation mean?

"Time average" means the average value over time (and the <> notation just means average, or expectation value) …

in this case, theoretically you'd integrate over a whole period, and then divide by the period.

However, in this case it's a bit obvious that the average of a cos or sin over the period is 0, so that just leaves you with … ? :wink:
 
  • #3
140
0
Ah, yes. Integrating over the period and dividing by the period gives you the correct answer. The cosines and sines vanish.

However, what exactly is this procedure though? I guess I've never seen integration related to an average.
 
  • #4
tiny-tim
Science Advisor
Homework Helper
25,836
251
However, what exactly is this procedure though? I guess I've never seen integration related to an average.

uhh? :redface: … how else would you define an average of something that varies continuously? :smile:

(and how do you think root-mean-square voltage etc is defined? :wink:)
 
  • #5
140
0
After a quick Google search I found some explanation. I have never seen continuous averaging before.

I have no idea what RMS voltage is either -- I'm a math major dabbling in physics.

Thank you for the help.
 

Related Threads on Thornton and Marion, Chapter 2, Prob. 22

Replies
8
Views
1K
Replies
5
Views
12K
  • Last Post
Replies
5
Views
3K
Replies
4
Views
1K
Replies
7
Views
5K
Replies
11
Views
2K
Replies
1
Views
491
Replies
7
Views
1K
Replies
4
Views
352
  • Last Post
Replies
2
Views
1K
Top