Does induced emf *always* have the same direction as the induced current?
Short answer: No.
Longer answer: As far as I can tell, the whole notion of
"induced current" is ill-defined.
-- I know what a voltage is.
-- I know what a current is.
-- I know what an induced voltage is, because there is a
Maxwell equation that quantifies the induced voltage.
-- I'm not exactly sure what an "induced current" is.
I say that even though googling "induced current" turns up
_more_ hits than "induced voltage" does.
Unfortunately, googling "induced current" turns up statements
like this:
«By Faraday's Law: the induced current is proportional
to the rate of the change of flux in a loop of wire.»
Alas I cannot accept such a definition. Faraday's law, which
is also part of the Maxwell equations, says that the _voltage_
is proportional to the rate of change of flux. If you cannot
tell the difference between voltage and current, your license
to define things is revoked.
As far as I can tell, to the extent that "induced current"
means anything it all, it refers to the current somehow
«associated» with the induced voltage. I put «associated»
in scare quotes, explicitly intending it to be a loose,
non-technical, non-quantitative notion.
In contrast, quantitatively, i.e. as a corollary of the
Maxwell equations, we can write
V = L dI/dt
whenever L is well defined, since flux = L I (which I take
to be the definition of inductance).
Given an inductor: for a sinusoidal signal, or for any
Fourier component of a more complex signal, we can write:
I = sin(ω t) "induced" current
V = L ω cos(ω t) «associated» voltage
This is a simple, direct consequence of the well-established
laws of physics. This suffices to prove that the current and
voltage do not "always" have the same direction.
======================
BTW FWIW: The term "emf" is passé ... sorta like electrical
"condenser", only worse. Previous discussions in this forum
have failed to reach a consensus on what "emf" meant. It's
ambiguous. That doesn't bother me, since I never use the
term anyway. We can avoid the ambiguity by using unambiguous
replacement terms. Depending on context, "emf" translates to:
-- the plain old voltage
-- the Thévenin equivalent open-circuit voltage