Angle of Deflection of an electron in an magnetic field

[Lunar_Lander]

Homework Statement

Electrons first run through an acceleration voltage of U = 25 kV before entering an homogenous magnetic field which is perpendicular to the electron beam (B=6*10^-3 T). The starting velocity of the electrons shall be zero.

What is the deflection angle α, if the magnetic field has a length of 5 cm (This means that the magnetic field stretches out for 5 cm in front of the electron accelerator)?

Homework Equations

0.5*m*v²=e*U can be transformed to give the velocity of the electrons.

q*(v x B) is the Lorentz force. (Or also: F_L = q*v*B)

The Attempt at a Solution

From the first equation I know that the velocity of the electron is 9.376*10⁷ m/s. That gives a Lorentz force of 9.001*10^-14 N. But how can I get the angle? I read somewhere that the tangent could be important, but I don't know how :(.

 

[gneill]

What is the shape of the trajectory of a charged particle moving perpendicularly to a magnetic field? (Hint: in what direction does the force act on the particle?)

 

[Lunar_Lander]

If you inject a charged particle into an magnetic field that is perpendicular to it, it will go around in a circle, with the Lorentz Force pointing "inside" the circle.

And in this case, we only get an arc and not a circle because the magnetic field is too "short" to make it go into a circle.

But I am still missing out the final idea on how to solve it, I can imagine some kind of triangle that you can form with one side being the lenghth of the B-Field and the hypotenuse being the arc. Or is that completely wrong?

 

[gneill]

If you inject a charged particle into an magnetic field that is perpendicular to it, it will go around in a circle, with the Lorentz Force pointing "inside" the circle.

And in this case, we only get an arc and not a circle because the magnetic field is too "short" to make it go into a circle.

But I am still missing out the final idea on how to solve it, I can imagine some kind of triangle that you can form with one side being the lenghth of the B-Field and the hypotenuse being the arc. Or is that completely wrong?

You'll want to determine the radius of the arc that the electron will follow. Take the center of that circle (which the arc is part of) as the center of an x-y coordinate system. You should be able to determine the point where the arc and boundaries of the magnetic field intersect. More importantly, ponder on the angle between the two radius vectors: 1) when the electron first enters the field, and 2) where the electron just leaves the field.

 

[Lunar_Lander]

I found it, thanks :)!