Why is the E cross B drift velocity constant in charged particle motion?

In summary, the problem is trying to find the velocity when there is a magnetic field and the charge has a component of its velocity in the same direction as the magnetic field.
  • #1
alannakamura
4
0

Homework Statement



good afternoon,

I am a problem with the E cross B drift velocity calculus from a charged particle. At first, an eletron is submitted a uniform magnetic field and have a circular trajetory. After, appear a uniform eletric field that create a drift velocity. The equations are:

m dv/dt = qE + qv X B.

Taking v constant or the mean aceleration zero, arrive the equation:

v = (E X B)/B^2

But I don't understand why the v is constant or mean aceleration is zero. Anyone can help me?
 
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  • #2
I think you first have to realize that you are dealing with vector quantities.

With just a magnetic field of strength B you have the force acting on a charge q moving with velocity v is mdv/dt = q vXB (where vectors are bold face).

In this situation if you assume that the charge moves perpendicular to the magnetic field (what does that mean for the term vXB?) then, as you have stated, the charge moves in a circular path.

Question: What if the charge of the particle has a component of its velocity in the same direction as the magnetic field?

When you apply an electric field, E, the total force acting on the charge is mdv/dt = qE + q vXB.

The velocity you determine (making use of vector notation) v = EXB/(B.B) is the condition for dv/dt = 0 and its not clear to me whether you have presented all the information?
 
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  • #3
sorry for I don't put in vector notation, but I write thinking in vectors. The problem is why dv/dt=0?
 
  • #4
No problem about the vector notation.

The reason I wondered if there was more to the question because, in general, dv/dt wouldn't be zero.

Look up "velocity selector", for example https://en.wikipedia.org/wiki/Wien_filter (must admit I learned something as well, I never realized that velocity selectors in this case were referred to as Wien filters).

If you look at the attached figure (taken from Wikipedia) you can see that the choice v = EXB/(B.B) is quite special.

velocity selector.png
 
  • #5
I think nobody erason for dv/dt=0 too. But, in all places that I read they use this condition. Is it only a hipothesis? I want to know if it was a hypothesis or something general of the problem.
 
  • #6
Something else you need to consider: To derive the expression for v you used the result
vXB = - E
then I guess you took the cross product with B giving
BX(vXB) = -BXE = EXB
Using the rules for vector cross product gives the left hand side as
v B2 + B v.B
so that you can only determine v for the case v.B = 0, namely that v and B are perpendicular
 
  • #7
v is the perpendicular component of velocity related tho magnetic field.
 

FAQ: Why is the E cross B drift velocity constant in charged particle motion?

1. What is E cross drift calculus?

E cross drift calculus is a mathematical tool used to study the behavior of charged particles in electric and magnetic fields. It involves calculating the trajectory of a particle as it moves through these fields and determining its velocity and acceleration at any given point.

2. How is E cross drift calculus used in physics?

E cross drift calculus is used in many areas of physics, including plasma physics, astrophysics, and particle physics. It is particularly useful in understanding the behavior of charged particles in the Earth's magnetic field, as well as in the design of particle accelerators and plasma confinement devices.

3. What are the key equations in E cross drift calculus?

The primary equations used in E cross drift calculus are the Lorentz force equation, which describes the force on a charged particle in an electric and magnetic field, and the equation of motion, which describes the particle's trajectory. Other important equations include the equations for magnetic and electric fields, as well as the equations for the particle's velocity and acceleration.

4. What are some real-world applications of E cross drift calculus?

E cross drift calculus has many practical applications, such as in the development of particle accelerators for medical and scientific research, as well as in the study of space weather and its effects on satellites and spacecraft. It is also used in the design of electric motors and generators, as well as in the study of plasma physics and fusion energy.

5. What are some limitations of E cross drift calculus?

E cross drift calculus is based on simplifying assumptions and idealized conditions, so its predictions may not always accurately reflect real-world situations. Additionally, it may not be applicable to extremely complex or chaotic systems where the behavior of charged particles cannot be easily predicted. It also does not take into account relativistic effects, so it may not be accurate at very high speeds or energies.

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