Lorentz Foce,conservative?

[Kolahal Bhattacharya]

Lorentz force is F=q(E+v cross .................................................. .................(1)
We seem to be interested only in B[=A(q/r^2)(v' cross ].What about E?Is it an electrostatic field?I suppose not.If not,then should be time dependent and del cross E=-(d/dt)B
Taking line integral of (1),W=integral(a to F.dr
=integral(a to E.dr + 0
Does this mean F(mag) is conservative?
if a and b are the same,Will W=0?
in that case will Lorentz force be conservative?
however, i saw in Griffiths's Quantum Mechanics that magnetic forces cannot be expressed like (-dV/dx) like other conservative forces.
what is the physics?

[Kolahal Bhattacharya]

please dont mind the b's are missing.

[dextercioby]

The Lorentz force

\vec{F}=q\vec{E}+q\vec{v}\times\vec{B} (1)

generalizes the magnetostatics and the electrostatics forces, with the latter being the Coulomb force. If the fields depend on time, which means the electrostatic and magnetostatic regimes are not valid anymore, then (1) is not conservative. It doesn't derive from any potential. But in the electrostatics, it's well known that the Coulomb force is conservative, since it derives from the Coulomb potential.

So the main idea is "time-dependence of fields".

Daniel.

[Kolahal Bhattacharya]

Thank you,daniel.I got your point.
But still unclear is that q(v cross B) part.Anyway F(mag)=q(v cross B) is zero even if the field is time dependent.So,can it be conservative?Actually,I am not sure that curl of B(t)=0 in that case.
what about the QM book confusion?Why magnetic forces cannot be expressed as F=-dV/dx like other conservative forces.