# Particle in an Electric Field

1. Jan 18, 2009

### Gogsey

A proton is initially (at time zero) at rest at the origin when a uniform, sinusoidally-varying electric field is turned on, exerting a force F(t) = Fo sinwt on the proton. Calculate the velocity and position as functions of time. Show that the proton moves farther and farther from the origin as time goes on. Calculate the numerical value of the average velocity (averaged over a whole number of cycles) if Fo = eE , w = 2 pi f , with E0 = 5×104 volts/metre, f = 20 MHz.

Firstly there is no velocity term in the equation so I was a little confused, but tried integrating mdv = Fo sinwt dt, and got, V = -Fo coswt/wm, and then again to get x = -Fo sin wt/w^2m.

Can anyone confirm this is right nor not. And if not, what am I doing wrong?

As you can tell, my mechanics is extremely rusty. Thats cause I haven't done any in 2 years, and there was no review or older material at the beginning of this course.

Thanks

2. Jan 19, 2009

### gabbagabbahey

Shouldn't there be a constant of integration or two? Given that v(0)=x(0)=0, what must those constants be?

3. Jan 19, 2009

### Gogsey

Well, those constants should be Vo and Xo, but both are zero at time = 0, so they don't contribute to the question at all.

4. Jan 19, 2009

### gabbagabbahey

Nope, you definitely need to consider the constants of integration. Without them you have:

$$v(t)=\frac{-F_o}{m\omega}\cos\omega t \implies v(0)=\frac{-F_o}{m\omega}\cos(0)=\frac{-F_o}{m\omega}\neq 0$$

and x(t) simply oscillates instead of increasing as expected.

So clearly something is wrong

5. Jan 19, 2009

### Gogsey

Is this anything to do with phase angles? Like cos(wt + phi), and that sort of stuff?

6. Jan 19, 2009

### Gogsey

Can you find the constants of integration by solving the initial value conditions? This seems like the logical thing to do.

7. Jan 19, 2009

### gabbagabbahey

Yes, that's exactly what you need to do. Give it a shot and see what you get.

8. Jan 19, 2009

### Gogsey

Ok, so the first one should be Fo/mw. The next ones a little trickier. First attempt I got, the constant to be zero for the seond integral.

9. Jan 19, 2009

### gabbagabbahey

Good

Better show me your attempt at that one

10. Jan 19, 2009

### Gogsey

Ok,

So for v(t)=-Fo sinw(t)/mw + Fo/mw.

Actually, th prof skipped the integration for finding X, although it was an equation without any constant, ie, this equation had a cos function, so there was no constant in this part. It was simply dv/dt=Fo sinw(t), and he skipped the integration, to find x. So I don't really know how to do this.

All I can think of is that it could be x=Fo sinw(t)/mw^2 + Fot/mw + C. And since t is zero,
C=0.

Last edited: Jan 19, 2009
11. Jan 19, 2009

### gabbagabbahey

You mean x(t) right?

$$v(t)=\frac{-F_0}{m\omega}\cos\omega t +\frac{F_0}{m\omega} \implies x(t)=\int v(t)dt=\frac{-F_0}{m\omega}\sin\omega t +\frac{F_0 t}{m\omega}+C$$

When you substitute the initial conditions, you should find that $C=0$.

What happens to x(t) as $t \to \infty$?

To average some periodic function f(t) over a period T, you use the formula $$f_{ave}=\frac{1}{T}\int_0^T f(t)dt$$

Apply that to v(t) to find your average velocity.

12. Jan 19, 2009

### Gogsey

Sorry, a bit lost now, I haven't seen this before, at least that I can remeber. Definatley not seen it in my 2nd year Mechanics course.

13. Jan 19, 2009

### gabbagabbahey

You should have covered time averages in Calc I or II. Try applying the formula and see what you get.

14. Jan 19, 2009

### Gogsey

Does this mean that I inetegrate v(t), and then after the integration I get what I had before times 1/t, then c = this integral evaluated at t(cause at zero its zero), times T?

15. Jan 19, 2009

### gabbagabbahey

This integral has nothing to do with determining $C$. You already showed that $C=0$.

This integral is what you need to compute in order to find the average velocity and answer this part of your question:

In the formula I gave above, capital T represents the period of v(t).

$$T=\frac{1}{f}=\frac{2\pi}{\omega}$$

16. Jan 20, 2009

### Gogsey

This may sound stupid, and it shows how little I know about this, but when you find Vav, which turns out to be just Xo (fromT to 0) * 1/T, which is then is just x(T) * 1/T.

Do you sub in T for the sin wt part, ie for the w(t)?

I'm pretty sure you don't but I wouldn't want to make this silly mistake.

17. Jan 20, 2009

### gabbagabbahey

Of course. To find the value of x(T), you substitute t=T everywhere into the expression for x(t).

18. Jan 20, 2009

### Gogsey

Maybe I said that wrong.

Vav = 1/T (Fosinwt/mw^2 + FoT/mw)

So 1/T cancles with the second term T to get Fo/mw.

Now should the first term be Fo sin wt/Tmw^2, or Fo sin wT/TmW^2?

19. Jan 20, 2009

### gabbagabbahey

Fo sin wT/TmW^2

for the exact reason stated in my previous post.

20. Jan 20, 2009

### Gogsey

Awesome., thanks. And wT still is 2 pi f, which is given in the equation, right? By going from wt to wT, nothing has changed in this sense.