Here is my appreciation of the interaction of a B field with an emf driven
current in a wire:
General notation: [x] specifies that x is a vector.
Let [i], [j] and [k] be the unit x,y,z vectors : to the right, up the page,
and out of the page, respectively.
The wire lies along the x axis with the current consisting of positive
carriers moving to the right with a
drift velocity [v] = v[i]. The magnetic field is everywhere [B]
= -B[k]:into the page. The qVxB force gives the carriers an additional
velocity component u[j], up the page. Thus, the (positive) carriers in the
wire have a resultant velocity [w] = v[i] + u[j], with components to the
right, and up the page.
This resultant velocity is in the first quadrant of the xy plane making an
angle TH with the y axis, where Tan(TH)=v/u. Note that TH is also the angle
between the total magnetic force [w]x[B] and the y axis [j].
The rate at which the magnetic force does work on a charge carrier is:
P = q*( [w]x[B] ) DOT [w] = q*( [w]x[B] ) DOT ( v[i] + u[j] )
Performing the DOT product:
P = q* | [w]x[B] | * (-v*Cos(TH) + u*Sin(TH) )
This is ostensibly zero, because v/u = tan(TH), by construction. The
magnetic force does no net work, but acts as a "go-between" to enable the
external emf agent to do the work.
The first term is power taken from the external agent; the second term is
power given to the current carriers. One might say that the magnetic field
delivers energy to the [j] motion of the carriers, but it gets that energy
from the [i] motion of the carriers - which energy ultimately comes from
whatever emf is driving the [i] current.