How Do You Solve Relativistic Dynamics Problems Involving Lorentz Force?

[sol66]

Homework Statement

Assume that in all inertial frmaes the force on a charged particle is gen in the usual Lorentz force law:

F = dp/dt = q(E + V x B) **(p is the relativistic 3 vector momentum)

Components of the four-force needs to be expressed in terms of E and B.

Homework Equations

F = ($$\gamma$$*F*V, $$\gamma$$*q(E + V x B) "3 vector portion")

4 Vector Momentum = P = m$$\gamma$$, m$$\gamma$$v "3 vector portion")

The Attempt at a Solution

So when attempting to solve this problem, I took the derivative of the relativistic 3 momentum vector and got ...

(dv/dt)(m$$\gamma$$ + $$\gamma$$^3v^2] = q(E + V x B)

My problem is that after look at this equation, I've realized that it is too difficult to solve for v in terms of E and B ... so there must be something I'm doing wrong. If I could solve for v then I could just simply plug it in my four vector formulas for force and velocity. I need to find the components for those vectors. Any idea how to fix this? Thanks.

 

[mark726]

Your approach is on the right track, but there are a few things that need to be addressed. First, the derivative of the 3-momentum vector should be with respect to proper time, not coordinate time. This will give you the correct expression for the 4-force.

Secondly, in your equation, you have mixed up the 3-momentum vector and the 4-momentum vector. The 3-momentum vector is given by p = m\gamma\vec{v}, while the 4-momentum vector is given by P = (m\gamma, m\gamma\vec{v}). The 4-vector equation for the force should be F = dp/d\tau = q(E + \vec{v} \times B).

Finally, to express the components of the 4-force in terms of E and B, you can use the Lorentz transformation to transform the electric and magnetic fields from one frame to another. This will give you the correct expression for the 4-force in terms of the electric and magnetic fields.

I hope this helps! Let me know if you have any further questions.