A particle's momentum in a magnetic field

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To calculate a particle's momentum in a magnetic field, the centripetal acceleration formula is applied, linking velocity, radius, and Lorentz force. The momentum is derived using the relationship between the radius of circular motion and the velocity components, particularly focusing on the component perpendicular to the magnetic field. The final expression for momentum is given as p = (Blqsinα)/(2π), emphasizing the importance of the angle and the magnetic field strength. There is a discussion about whether to consider the sum of velocity vectors, as momentum has components in both directions. The calculations presented seem valid, and the approach is generally accepted.
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Homework Statement
A particle with a charge equal to that of an electron enters a magnetic field. The direction of the particle's velocity
makes an angle of 45 degrees with the magnetic field lines. The particle moves in a spiral with a pitch of
##l##=2 cm. What is the magnitude of the particle's momentum when the magnetic field induction B is 0.02 T ?
Relevant Equations
##F_{Lorenz}=qvBsin\alpha##
##F_{centrifugal}=\frac{mv^{2}}{R}##
In this case:
##F_{Lorenz}=F_{centrifugal}##
to calculate momentum (p), do I need to use sum of speed vectors? Maybe someone can help me to solve this problem. For now, my solution looks like this:

$$a_{centrifugal}=\frac{v^2}{R}=\frac{F_{Lorenz}}{m}=\frac{qvBsin\alpha}{m}$$
$$\frac{v}{R}=\frac{qBsin\alpha}{m}\implies m=\frac{qRBsin\alpha}{v}$$
then getting R (T here is time period):
while particle is spinning around magnetic field lines (around x) and going to the side (direction x), the circumstance and ##l## are done in the same period of time (T)
$$T_{1}=T_{2}$$
$$\frac{l}{v_{x}}=\frac{2\pi R}{v}$$
$$v_{x}=vcos\alpha$$
Then R:
$$R=\frac{l}{2\pi cos\alpha}$$
then getting back and putting R there:
$$p=mv_{x}=mvcos\alpha=\frac{qBRsin\alpha}{v}\cdot vcos\alpha=\frac{qBlsin\alpha}{2\pi cos\alpha}\cdot cos\alpha=\frac{Blqsin\alpha}{2\pi}$$
$$p=\frac{Blqsin\alpha}{2\pi}$$
I am not sure about momentum being calculated like that, should it be sum of both vectors maybe, or how it works? can someone explain to me, or solve this problem? Thanks.
 
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The velocity of the circular motion is not the full velocity but just the component perpendicular to the field. And yes, the momentum have components on both directions.
 
nasu said:
The velocity of the circular motion is not the full velocity but just the component perpendicular to the field. And yes, the momentum have components on both directions.
I have came up with different solution. Could you check it aswell? Do not mind language. I'm adding image.
 

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Antoha1 said:
I have came up with different solution. Could you check it aswell? Do not mind language. I'm adding image.
1744840280005.png

This looks good to me.
 
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