Trouble understanding this hw problem

[huk]

An electron is moving in a plane that also contains a long straight current-carrying wire. The electron is heading at a 45° angle toward the wire, with a speed of 3.8 106 m/s, when it is 43 cm away. The electron reaches only as close as 1 cm, before being repelled away, always moving in the same plane. What is the current in the wire?

I could picture it well but i have no idea how to solve this problem. All i knwo is that its velocity should be constant because its in the plane perpendicular to the magnetic field and the force i think.. Can anyone tell me how to do it or give me any hints that would help me think better.
Your help would be greatly appreciated!
Huk

 

[mmwave]

I could picture it well but i have no idea how to solve this problem. All i knwo is that its velocity should be constant because its in the plane perpendicular to the magnetic field and the force i think.. Huk

First, have you drawn a picture yet? Second, be careful in your choice of terms. Is it really the velocity that is constant? Review the force law for moving charges and think about what is required for uniform circular motion. I think these hints will change your thinking.

 

[OlderDan]

An electron is moving in a plane that also contains a long straight current-carrying wire. The electron is heading at a 45° angle toward the wire, with a speed of 3.8 106 m/s, when it is 43 cm away. The electron reaches only as close as 1 cm, before being repelled away, always moving in the same plane. What is the current in the wire?

I could picture it well but i have no idea how to solve this problem. All i knwo is that its velocity should be constant because its in the plane perpendicular to the magnetic field and the force i think.. Can anyone tell me how to do it or give me any hints that would help me think better.
Your help would be greatly appreciated!
Huk

I can't see any easy way to do this. I'm not sure this will work out, but the only thing that comes to mind so far is to use the constant speed condition to find the force as a function of the distance from the wire in terms of the unknown current, and the known relationship between field strength and distance for a long straight wire. At any distance from the wire the electron's path will have a radius of curvature determined by that force and the speed (centripetal force). Perhaps the relationship between radius of curvature and coordinate derivatives combined with the constant velocity will lead you to the answer

http://mathworld.wolfram.com/RadiusofCurvature.html

The parametric form of the radius (eq 4) seems to hold some promise. It looks to me like constant speed should lead to a constant numerator. Can you relate the acceleration components to the velocity components? If this leads you to an expression for the acceleration at any point (such as the point of closest approach) you should be able to then find the current.

 

[qbert]

This was a tricksy problem.

If we assume the current is lying along the y-axis and headed
in the y direction the B field from the wire is
$$\vec{B} = \frac{-\mu_0I}{2 \pi x} \hat{z}$$
We can take the coordinates so the particle has an initial position (x_0, 0).

The force on the particle is given by
$$\vec{F} = q \dot{\vec{x}} \times \vec{B} = m \ddot{\vec{x}}$$
working that out gives two coupled ODE
$$\ddot{x} = -\frac{c}{x} \dot{y} \; , \; \ddot{y} = \frac{c}{x} \dot{x}$$
where
$$c = \frac{q \mu_0 I}{2 \pi m}$$

Since the force is always perpendicular to the motion
the energy is conserved. (Mathematically you get
this by dividing the first by the second, cross multiplying
denominators and recognizing the derivative.)
so:
$$\dot{x}^2 + \dot{y}^2 = const = v^2$$
where v is the initial velocity.

We can solve this for y-dot, and plug into the x-equation.
$$\ddot{x} = \pm \frac{c}{x}\sqrt{v^2 - \dot{x}^2}$$

now we use the standard trick:
$$\ddot{x} = \frac{d}{dt} \dot{x} =\frac{d \dot{x}}{dx} \frac{dx}{dt} = \dot{x} \frac{d \dot{x}}{dx}$$

This leaves us a seperable equation. We can integrate that
using the initial conditions. To make things easier let's
decompose the velocity into x and y components so
$$\vec{v} = (-v \cos(\theta), v \sin(\theta))$$
The unstandard use of minus signs allows theta = 45 deg
for the original problem. (theta measures from the -x axis
o the y axis.)

then the integral gives us
$$\pm (\sqrt{v^2 - \dot{x}^2} - v\sin(\theta_0)) = c \ln{\frac{x}{x_0}}$$

This should hold for all time t, specifically, it should hold at the
closest point. but at the closest point x-dot is 0. and x is x_c.
(that is, the particle is moving parallel to the wire).

$$\pm v\left(1-\sin \theta_0 \right) = c \ln{\frac{x_c}{x_0}}$$

We can solve this for I.

giving
$$I = \pm \frac{2 \pi m v(1- \sin( \theta_0))}{q \mu_0 \ln{ \frac{x_c}{x_0}} }$$

We take the positive (by choice of the coordinate system I is in
the +y direction I>0). That means we had to take the negative root
clear back in the beginning.

Now we're done.

for the original problem putting in the mass/charge of an
electron and all the pertinent info gives
I = 8.43e-6 amps
using that I and all the problem info, I had maple do a DEplot
of the solution of the ODE system.
The attachment shows the graph from t=0.18 to 0.20 seconds.

 

[qbert]

oops. i didn't mean to do the homework problem
for the op. (i got carried away.)

 

[OlderDan]

]
We take the positive (by choice of the coordinate system I is in
the +y direction I>0). That means we had to take the negative root
clear back in the beginning.

Well done. Did you explore a second solution? For the same initial velocity, if the current is in the other direction the electron's path will curve toward the wire. It will eventually get back to the problem you solved, but with a smaller "initial" distance from the wire when at 45 degrees again. A larger current should still result in the 1cm closest approach.

I'm curious about the full trajectory of your solution. Initially I thought the path might be parabolic, but the possibility of a second solution makes me think otherwise. With a force that is always perpendicular to the velocity, diminshing as the distance from the wire increases, I don't see any possibility of multiple approaches to the wire, but I suspect that the y component of the velocity can retrograde one time for an approach, something along the lines of this diagram. Does something like this fit the solution?

 

[qbert]

yeah, i think the solutions are more loopy than parabolic,
which makes sense. it's easier just to play around with
the DE system using a nice numerical solve/plot package.

changing conditions from the original problem to
vx_0 = -3*cos(Pi/4)
vy_0 = 3*cos(Pi/4)
x_0 = 0.5
y_0 = 0

I = 1e-4 gives the first graph (t=0..3)
[inverting the equation for I = f(x_c) i plotted x_c given
the I, it's the red vertical line.]

and I = -1e-4 gives the second graph (t=0..1)
[this is weird, the two solutions for x_c are plotted,
but the negative solution gives nothing, and the positive
solution gives the farthest point. I'm not really sure
what's happened here. ]

notice the loops are tighter for a current down the wire
and the direction of motion changes.

if we go with a positive current (which is the only
one i trust.) and increase it. the solutions become
more and more circle like. this makes sense because
the field is better and better approximated by a const
linear field.

the third drawing is for i = 1amp. t = 0..0.001 (or something,
i forgot to write the time down before i exported it.)

 

[OlderDan]

yeah, i think the solutions are more loopy than parabolic,
which makes sense. it's easier just to play around with
the DE system using a nice numerical solve/plot package.

I don't have any tools handy to do numerical solutions. Apparently I was wrong about the electron not making multiple passes near the wire. It makes sense now that I think about it again. All that is required is that the radius of curvature be proportional to the distance from the wire. Your diagrams appear to have a smaller radius at the maximum distance than at some closer point, but I'm quite sure that is just a scaling issue. Any chance you could draw those so that the axes are to the same scale to better represent the shape of the trajectories?