Is B just that part of E that does no work?

B and E are two fields that act on charge, with the former doing no
work. So isn't B just that part of E which does no work? i.e. that
part of E which is normal to the velocity, v, of a charge and so could
be defined as E x v? So every E will have a B that is dependent upon
the path taken by a charge through it.

I want to calculate B for a straight wire using the above idea. I will
calculate the delayed E at a point due to the moving electrons, and
subtract that from E due to the stationary ions which should give zero
since a stationary charge experiences no force in practise. The
problem is why a moving charge, Q, experiences a force if E is zero
there? I suspect it's because it's E will have an effect on the
electrons in the wire because of the finite time it takes to effect
them, which in turn generate an E that effects Q so that an E normal
to v of Q is the net result.

Is this going to be a waste of time, so that I will end up with
something that conflicts with Maxwell's equations?

Thanks in advance.

.

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