# Lorentz transformations combined with force

• daselocution
In summary, the conversation discusses the relationship between Newton's second law and the acceleration of a particle in a magnetic field, as well as the use of this result in determining the radius of a particle's circular path in a magnetic field. It is noted that as the speed increases in a cyclotron, the radius also increases, but at relativistic speeds, synchronization issues arise and limit the amount of energy that can be given to particles.
daselocution

## Homework Statement

First part of the problem:
Newton’s second law is given by F=dp/dt. If the force
is always perpendicular to the velocity, show that F=gamma*m*a, where a is the acceleration.

Second part of the problem: Use the result of the previous problem to show that
the radius of a particle’s circular path having charge q traveling with speed v in a magnetic field perpendicular to the particle’s path is r = p/qB. What happens to the radius as the speed increases as in a cyclotron?

## Homework Equations

p=gamma*m*v

Fmagnetic field = qv x B = (in this case b/c of θ=90º) qvB

## The Attempt at a Solution

The first part: I am thinking that since the force is perpendicular to the path of motion, that the speed of the particle will not change, only it's direction--is this logical? If this were indeed the case, then I would solve as follows (and get the answer as directed):

F=dp/dt=d(m*gamma*v)/dt = m*gamma*d(v)/dt
=m*gamma*a

where dv/dt=a and where speed is unchanging so gamma should be constant

Second part of the problem:

F=m*gamma*acceleration=qvB

At this point I'm entirely unsure of how to proceed. I remember that in classical physics a=v2/r, but I don't know if that applies here.

If it does indeed apply here, then the answer seems to be straightforward:

F=m*gamma*acceleration=m*gamma*v2/r=qvB

r=m*v*gamma/(qb)=p/(qB)

Thus, as the speed increases in a cyclotron, the radius should increase as well

I would agree with what you have said. However you are perhaps expected to say something more about the cyclotron.

Even at non-relativistic speeds r increases with v. However r is directly proportional to v and since time for 1 rev=2∏r/v the frequency of revolution is a constant which makes accelerating particles easy while they are only traveling at about 0.1c or less. (apply an alternating electric field of the expected frequency)

However once relativistic speeds are approached r increases faster than v and the frequency of rotation reduces giving synchronization problems. Thus cyclotrons are limited in how much energy they can give particles especially light ones that go relativistic easily such as the electron. (There have been attempts to overcome this problem such as the synchrocyclotron)

Thank you very much--I went to office hours today to ask my teacher and he echoed more or less what you said about the speed of the cyclotron

## 1. What are Lorentz transformations in relation to force?

Lorentz transformations are mathematical equations that describe how space and time coordinates change between two different reference frames in special relativity. In the context of force, these transformations allow us to understand how objects with mass behave and interact with each other in different frames of reference.

## 2. How are Lorentz transformations used to calculate force?

Lorentz transformations include the concept of relativistic mass, which accounts for the increase in an object's mass as it approaches the speed of light. By incorporating this into the equations for force (F = ma), we can accurately calculate the force acting on an object in different reference frames.

## 3. Can Lorentz transformations change the direction of a force?

No, Lorentz transformations do not change the direction of a force. Force is a vector quantity, meaning it has both magnitude and direction. While Lorentz transformations can change the values of certain variables in the equations for force, they do not alter its direction.

## 4. How do Lorentz transformations affect the laws of motion?

Lorentz transformations do not affect the laws of motion themselves (such as Newton's laws), but they provide a more accurate and consistent way to apply these laws in different reference frames. They also help explain phenomena such as time dilation and length contraction, which are crucial in understanding the behavior of objects at high speeds.

## 5. Are Lorentz transformations only applicable to objects moving at relativistic speeds?

No, Lorentz transformations can be used for any objects moving at any speed, but their effects become more significant as an object approaches the speed of light. At slower speeds, the differences between classical and relativistic physics are negligible, but as an object's velocity increases, the need for Lorentz transformations becomes more apparent.

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