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I would really appreciate some help on the relativistic Lorentz force as applied to a charged particle; it's my first post on this site btw. Thanks in advance ;)

QUESTIONS

(a) A charged particle with total energy E0 is injected into a region of space containing a uniform electric field at right-angles to its direction of motion. It is deflected by the electric field and after a time is travelling at 45 degrees to its original line of flight. Show that the energy, E1, of the particle at that point is given by (where m is the mass of the particle):

E1^2 = 2*E0^2 - m^2*c^4

(b) The component of the particle’s velocity in the direction of its original line of flight decreases with time, even though no force acts upon the particle in that direction? Explain why this is so. Obtain an expression for this velocity component at the point where the particle is moving at 45 degrees to the original line of flight.

NOTATION

gamma(u) is the Lorentz factor for velocity u

I have taken the particle to be travelling initially in the +z direction at velocity u0, with the electric field acting in the +x direction. At 45 degrees the particle travels at a velocity u.

MY ATTEMPTED SOLUTION FOR (a)

Use the the energy momentum invariant:

E0^2 = {gamma(u0)*u0*mc}^2 + {mc^2}^2

E1^2 = {gamma(u)*u*mc}^2 + {mc^2}^2

E1^2 = {gamma(u)*mc}^2*{u0^2(1+cot^2(45))} + {mc^2}^2

since the z - velocity is unchanged in the IRF??

E1^2 = 2 * {gamma(u)*u0*mc}^2 + {mc^2}^2

The stated answer follows from algebra

MY ATTEMPTED SOLUTION FOR (b)

The momentum 4-vector for the particle at 45 degrees is:

P = (E1, px', py', pz')

P = (E1, gamma(u)*m*u0, 0, gamma(u)*m*u0)

So the z velocity component is gamma(u)*m*u0 where u^2 = 2*u0^2

The z velocity is different despite the absence of a force in the z direction due to the E-field having a z-component in the particle's IRF??

The reason I'm really unsure of my answer is that it is unclear which frames I am supposed to be calculating all the forces in, and I'm confused as to the geometric relationship between the different velocities in them.

Thank you!

VR