The Loop Rule: A Consequence of Conservation of Energy

[jdavel]

What if you simplify the problem?

Imagine a square wire loop, ABCD, of some resistance R moving at constant velocity, parallel to its plane and perpendicular to a constant B field. Label the leading corners A and B, and the trailing corners C and D.

There will be a constant current in the loop (let it be in the ABCD direction). The current will be the same in all four legs (AB, BC, CD, DA) of the loop.

Along legs AB and CD (excluding the corners!) the voltage is constant, call it zero. The current is caused exclusively by Faraday's Law. If there were an electric potential, you'd get a greater current than Faraday's Law says you should, and you don't.

Along legs BC and DA the magnetic force on the electrons is perpendicular to the wire, so Faraday's Law plays no part. The only other thing that can make electrons move is Coulomb's Law. So from B to C and from D to A (excluding the corners) there has to be a continuous rise in electric potential (conventional current).

So going around the loop, the voltage is zero along AB. Then there's a discontinuous drop at B to some value less than zero. Then there's a continuous rise from B to C to a value greater than zero. Then the voltage drops discontinuously to zero at C and so on... At the end of DA, the voltage drops back to zero just as you go around the corner; you're back to where you started, and the voltage has the single value of zero.

So, Faraday and Kirchoff are both in tact and operating compatibly!

PS I had to think about this one for a while: What makes the electrons go down the potential discontinuity at B and D?

 

[rbj]

i'm going to regret getting back into this.

i posed a similar question about this way back (that was never taken on, until now):

what happens when you have a single circular closed loop of wire in the presence of a non-zero and changing magnetic field? what happens when you apply

\oint_{s} \mathbf{E} \cdot d\mathbf{s} = - \frac {d\Phi_{\mathbf{B}}} {dt} ?

that first integral, on the left hand side, is a voltage. do you get zero?

jdavel, i think you have a point, but there are some problems.

The current is caused exclusively by Faraday's Law.

first, it's a voltage that gets caused by Faraday's Law. and

The only other thing that can make electrons move is Coulomb's Law.

is not true in general (perhaps you mean it only in the context of legs BC and DA).

the third problem is, i am not sure if the loop is moving parallel to the plane with a constant B field coming out that there is any net

\frac {d\Phi_{\mathbf{B}}} {dt} .

fourth, if there is a current in the loop, even legs BC and DA will have a voltage drop (and in the same direction around the loop).

however, how about a loop of resistors (in a square or a triangle or octagon or whatever symmetrical shape you like) that is placed right beside a big solenoid with a changing current in the solenoid causing a changing B field (this is essentially a transformer)? the axis of the loop and solenoid are in-line.

we know there will be an infinite number of little (infinitesimal) phantom voltage sources distributed around the loop (these come from the \frac{dA}{dt} in Tide's version), all oriented in the same direction (say, clockwize) with their voltages teaming up (not cancelling). this and Ohm's Law will determine some non-zero current going around the loop.

now, here is where physicists (likeTide) and engineers (like me) look at this differently:

i say the circuit of resistors, as drawn on a piece of paper, is not adhering to KVL, but if the circuit is modified, to have all of those little voltages sources tossed in (usually as one big lumped source), then KVL is satisfied but those voltage sources are not components of the circuit that we started with. for that reason, it is a deviation.

i'll let Tide spell out his version.

 

[rbj]

i don't know why, but i thought i would consult the "authoritatve" wikipedia to see what it had to say on the subject. a lot of language is similar to what i was saying all the time (Faraday's Law, "fixing" KVL) with one important exception: perhaps i should have used the term "EMF" instead of "potential difference". perhaps this might help Tide and i to sing the same tune.

from http://en.wikipedia.org/wiki/Kirchhoff's_circuit_laws#Kirchhoff.27s_voltage_law

The directed sum of the electrical potential differences around a circuit must be zero.

(Otherwise, it would be possible to build a perpetual motion machine that passed a current in a circle around the circuit.)

This law has a subtlety in its interpretation, because in the presence of a changing magnetic field the electric field is not conservative and it cannot therefore define a pure scalar potential—the line integral of the electric field around the circuit is not zero. Equivalently, energy is being transferred from the magnetic field to the current (or vice versa). In order to "fix" Kirchhoff's voltage law for this case, an effective potential drop, or electromotive force (emf), is associated with the inductance of the circuit, exactly equal to the amount by which the line integral of the electric field is not zero by Faraday's law of induction.

from http://en.wikipedia.org/wiki/Electromotive_force

Motional emf is ultimately due to the electrical effect of a changing magnetic field. In the presence of a changing magnetic field, the electric potential and hence the potential difference (commonly known as voltage) is undefined (see the former) — hence the need for distinct concepts of emf and potential difference. Technically, the emf is an effective potential difference included in a circuit to make Kirchhoff's voltage law valid: it is exactly the amount from Faraday's law of induction by which the line integral of the electric field around the circuit is not zero. The emf is then given by L di/dt, where i is the current and L is the inductance of the circuit.

Given this emf and the resistance of the circuit, the instantaneous current can be computed with Ohm's Law, for example, or more generally by solving the differential equations that arise out of Kirchhoff's laws.

 

[Tide]

rbj,

perhaps this might help Tide and i to sing the same tune.

Way to go, rb! :)