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Re: emf, potential, voltage

At 11:39 PM 3/11/01 -0500, Carl Mungan wrote:
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.

Exactly so.

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.

And that's exactly the explanation: proof by construction.

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.

I practically never use the term emf. When someone else uses it, I accept
it as a synonym for voltage.
-- I do not assume it is a _potential_ difference
-- Nor do I assume it is associated with a particular device; it could
refer to the voltage drop around an abstract loop somewhere in empty space.

Two obvious examples are batteries and ac generators. Resistors
always dissipate energy and hence are not emf's.

Again, usually the term emf is not usually associated with a device. I
don't think it is conventional to say that a battery "is" an emf. A
battery, or a resistor, could have an emf across its terminals. A separate
statement would be needed to describe the energy-source that produces the emf.

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?

Any attempt to distinguish capacitors from batteries is misguided. In my
book, a battery is just a high-value capacitor with lousy high-frequency
behavior and other nonidealities.

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:

They're not in a different category. Both are in the category of passive
reactive components.

both absorb energy in
certain parts of an ac cycle and release that energy in other parts
of the cycle.

Right.

Surely if I can talk about potential energy, then I can
talk about potential*?

Right, up to a possible overall constant of integration which produces some
non-uniqueness (i.e. gauge invariance).

Hence by definition, emf's appear to be
potential differences.

Unless there are stray time-varying magnetic fields, in which case we could
have a non-potential emf. Remember, I take emf to be synonymous with voltage.

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.

Hmmmm. "through" the device is problematical. Otherwise OK.

The condition of "no leakage" can (with a little refinement) be formalized
as one of the conditions necessary for the validity of Kirchhoff's laws.

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.

Right. This is a non-potential voltage situation for exactly the reasons
given.

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.

Right.

*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.

Right, again except for a possible overall constant of integration (the
gauge) that affects the numbers but does not affect the physics.