= Solution
The initial <momentum> magnitude satisfies $p^2/(m^2c^2)=\gamma_i^2-1=288$. Equality of longitudinal and transverse magnitudes therefore gives
$$
p_{z,i}=p_\perp=12mc.
$$
An electric field along $z$ changes $p_z$, while the magnetic <Lorentz force> only rotates the transverse momentum. Its magnitude $p_\perp$ remains fixed. At the longitudinal turning point $p_z=0$, hence
$$
\boxed{\gamma_f=\sqrt{1+\frac{p_\perp^2}{m^2c^2}}=\sqrt{145}=12.04159.}
$$
For an electron initially moving toward positive $z$, the decelerating field points toward positive $z$ because its charge is $-e$. The electric work over $\Delta z=1\,\mathrm m$ gives
$$
-eE_z\Delta z=(\gamma_f-17)mc^2,
\qquad
\boxed{E_z=2.53375\,\mathrm{MV\,m^{-1}}.}
$$
The radius is determined by transverse momentum rather than total <energy>:
$$
\boxed{R(z)=\frac{p_\perp}{e|B|}=\frac{12mc}{e|B|},}
$$
so the requested radius plot is horizontal. The cyclotron angular frequency changes as $e|B|/(\gamma m)$ and the helix pitch shrinks to zero. These statements neglect radiation, as stipulated.
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