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emf, potential, voltage



I would like to sharpen my understanding of the distinction between
potential difference, voltage, and emf. All three of these quantities
are measured in volts, and texts seldom are sufficiently clear about
the differences. I briefly summarize my thinking below and invite
corrections and clarifications.

1. All potential differences are voltages, but not all voltages are
potential differences.

For example, the line integral of the induced electric field in a
closed loop in a region of time-varying magnetic field is a voltage
but not a potential difference. This arises because the electric
field is not conservative and hence cannot be written as the gradient
of a potential*. If you go around the loop, the net voltage is not
zero.

2. We define a source of emf as a device or element that supplies
electromagnetic potential energy to charges flowing through it. The
emf is the potential energy change per unit charge.

Two obvious examples are batteries and ac generators. Resistors
always dissipate energy and hence are not emf's. Although capacitors
act like batteries, no charge flows through them, so is that why they
are not described as emf's even though they can supply energy, say to
a camera flash?

The formulas for inductors in textbooks describe the voltage across
their leads as being an emf. It is not clear to me why inductors are
in a different category than capacitors: both absorb energy in
certain parts of an ac cycle and release that energy in other parts
of the cycle.

More generally, I am not clear on when emf's are potential
differences. Surely if I can talk about potential energy, then I can
talk about potential*? Hence by definition, emf's appear to be
potential differences. This seems to be okay for say a battery or
motional emf or an inductor. (Take the inductor to be an ideal toroid
if you are worried about flux leakage.) These all have two terminals
and each terminal has a well-defined voltage with respect to ground
(assuming for simplicity that the circuit is not floating). Any path
between these two terminals will give the same voltage: you can go
through the device, follow a path around the device, etc. However
it's clearly not okay for a ring in a time-varying magnetic field: I
cannot associate a specific voltage with each spot on the ring
because every time I go around the ring that value changes. What
happens if I make a tiny gap in the ring? This is still not akin to
motional emf, because if I attach voltmeter leads to the two end
points at the gap in the ring the voltage reading depends on the path
the leads take through the magnetic field region. It's obviously not
a potential difference therefore.

*Footnote: I am implicitly defining a potential difference as a
difference in potential between two points. By choosing one of these
points arbitrarily to be a reference, I can convert any potential
difference into a potential. Hence, the terms "potential" and
"potential difference" can be considered synonyms whenever desired.