How Does Halving the Magnetic Field Affect Particle Displacement?

[kaspis245]

Homework Statement

A particle with a charge enters an area where it becomes affected by resistance force, which is directly proportional to its velocity. The particle moves 10 cm in that area and stops. If there was a magnetic field in that area the particle with the same initial velocity could move a total displacement of 6 cm. What distance could the particle move if the magnetic field would be two times weaker?

Homework Equations

Laws of motion

The Attempt at a Solution

$F_r$ - the resistance force
$F_{rx}$ - horizontal resistance force
$F_{ry}$ - vertical resistance force
$F_m$ - force caused by the magnetic field
$v$ - initial speed

The first drawing shows particle moving in an area without magnetic field and the second one with it.

When the particle is moving in an area with magnetic field it is affected by horizontal forces $F_m$, $F_{rx}$ and one vertical force $F_{ry}$.

Now, I think that the particle in both diagrams should have the same vertical accelerations, however due to the fact that the particle in magnetic field moves only 6 cm it appears otherwise.Why is that so? Why does the particle in magnetic field has a smaller vertical displacement? What additional forces cause this? I can't see how $F_m$ can affect this since it is only involved in horizontal motion.

 

[TSny]

Now, I think that the particle in both diagrams should have the same vertical accelerations, however due to the fact that the particle in magnetic field moves only 6 cm it appears otherwise.Why is that so? Why does the particle in magnetic field has a smaller vertical displacement? What additional forces cause this? I can't see how $F_m$ can affect this since it is only involved in horizontal motion.

Is the magnetic force in the x direction throughout the motion or is it in the x direction only at the beginning of the motion?

 

[kaspis245]

Is the magnetic force in the x direction throughout the motion or is it in the x direction only at the beginning of the motion?

The magnetic field is present throughout the motion.

 

[TSny]

But does the magnetic force always have the same direction throughout the motion?

 

[kaspis245]

But does the magnetic force always have the same direction throughout the motion?

No, I suppose it would not. At some point it would have a vertical component which would cause the difference in vertical accelerations.

 

[TSny]

No, I suppose it would not. At some point it would have a vertical component which would cause the difference in vertical accelerations.

Right.

 

[kaspis245]

I need some help. How can I describe particle's motion in the magnetic field?

Particle's motion in the the area without magnetic field can be expressed like this:
$m\frac{dv}{dt}=-Kv$ where K is some constant.
$\frac{dv}{dt}=\frac{K}{m}v=-kv$
$t=-\frac{1}{k}lnv$
$v=e^{-tk}$

 

[TSny]

I assume that the magnetic field is uniform and perpendicular to the x-y plane. So, the trajectory of the particle is similar to a bubble chamber track as shown below.

This is a problem that you can solve by inspection of the differential equation for the motion. If $\mathbf{r}$ is the position vector of the particle, what is the differential equation for $\mathbf{r}$. That is, using Newton's 2nd law can you find an expression for $\ddot{\mathbf{r}}$ in terms of $\dot{\mathbf{r}}$, the mass, the damping constant, and the magnetic field?

EDIT: It might be best to write separate differential equations for the x and y components of $\mathbf{r}$ .
Also, I changed the figure below to correspond to the specific data of this problem.