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Re: [Phys-l] induced electric field

Bob -- You have exactly stated what has puzzled me about this. But I think the other posts in this thread are beginning to lead me out of this thicket.

I think everything in your development is OK up to and including the step where you find the current. The problem comes when we go to find the electric field associated with that EMF. If we want to say that its magnitude is the EMF/path length and its direction is tangential to the loop, we are then assuming that the electric field is the same magnitude at all points on your wire loop and tangential to the loop. In other words, you are assuming a symmetry that requires your loop to be at the "center" of some radially symmetric electric field. But suppose you cannot assume that the field has the same magnitude at all points on the loop -- say because it is that second loop you and I were both thinking of, near but not touching the first. At the point where the second loop is nearest the first, the electric field is pointing "the wrong way" - not in the direction of the current. But I think you could find the electric fields at every point on your second loop using the methodology from the first loop and then calculate Integral(e dot ds) to find the EMF for the second loop. I have not done this calculation, but I believe that miracle-like, it will come out as it should: dB/dt times area.

So to sum up what I think I've learned here: there is no contradiction. There is a unique electric field at every point in the plane where this problem takes place. The confusion did come from not realizing the importance of the symmetry when stating the problem. I'll add that the problem shows up regularly on the EM section of the AP physics C test but I don't recall symmetry or boundary conditions ever being mentioned.
________________________________________
From: phys-l-bounces@carnot.physics.buffalo.edu [phys-l-bounces@carnot.physics.buffalo.edu] On Behalf Of LaMontagne, Bob [RLAMONT@providence.edu]
Sent: Monday, November 23, 2009 6:15 PM
To: Forum for Physics Educators
Subject: Re: [Phys-l] induced electric field

Instead of the differential form of Maxwell's equations, I find the integral form a little easier to visualize here.

If I take your question as-is and don't turn it into a question that may be easier to answer, I have to conclude that the question is not answerable. If I form a circular loop in the x-y plane and integrate around that circle, I can easily find a value for Integral(E dot ds) = - d(B_flux)/dt. The result is dB/dt times the area of the circle. The EMF around this loop has this same value. If a circular loop of wire is placed in the same position as the loops we integrated over, the current through the wire loop will be the EMF/R (where R is the resistance of the loop). Since no boundaries are specified, I can move the loop to any position that I want in the x-y plane and I will get the same current in the same direction (clockwise or counterclockwise). If I now take two loops of wire and place then so their edges are almost just touching, the current in one is flowing in the opposite direction as in the other at the point where they almost touch. So if we are thinking in terms

of electric fields pushing an charge, the field obviously has two opposite directions at the same point - depending on which loop I am looking at. I take this absurdity to indicate that the question cannot be answered exactly as it is posed.

Bob at PC

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of Philip Keller
Sent: Monday, November 23, 2009 10:35 AM
To: 'Forum for Physics Educators'
Subject: Re: [Phys-l] induced electric field

I'm in over my head...but I want to re-ask my question, and then I can
work on the answer on my own:

"In a region in space where the magnetic field is steadily increasing
at 1 T/s in the z-direction. A proton is released from rest at the
point with (x,y,z) coordinates (1,0,0) . What is the magnitude and
direction of the force that the proton experiences as a result of this
changing field?"

Is this an answerable question as is or do I have to specify the
boundary conditions that give rise to the increasing magnetic field?
If it is answerable as is, what is the answer?! And why did the
coordinates of the location matter?

Sorry if I am being obtuse. It isn't willful.

-----Original Message-----
From: phys-l-bounces@carnot.physics.buffalo.edu [mailto:phys-l-
bounces@carnot.physics.buffalo.edu] On Behalf Of John Mallinckrodt
Sent: Monday, November 23, 2009 4:38 AM
To: Forum for Physics Educators
Subject: Re: [Phys-l] induced electric field

On Nov 22, 2009, at 8:36 PM, John Denker wrote:

I think the shoe is on the other foot. Some folks are
assuming without proof -- indeed without evidence -- that
the squarish solution exists.

... proponents of squarish solutions are encouraged
to exhibit some such thing, in specific and formal terms,
and show that it solves the relevant Maxwell equation in
the given situation. I don't think it exists: too much
curl near the corners of the "square" and not enough
elsewhere.

Well, in fact, I actually did do some numerical calculations based on
a Biot-Savart-like transformation of Faraday's law and they clearly
do support the "squarish solution."

It shouldn't surprise anyone that the solution we seek is identical
in form to that for the B-field produced by a uniform current density
within a region having a square cross section. For a quick and dirty
solution I set up a spreadsheet with a 20x20 array of 400 "wires"
carrying "current" in the +z direction and used it to calculate the
resulting field at arbitrary positions in the x-y plane. Inside the
square, the solution is a little too too susceptible to the varying
distance to the nearest wire to be reliable, but outside the solution
is quite stable and smooth. For a radius of 15 units, I calculated
the B-field every 3 degrees from 0 to 45 degrees and found that the
field is purely tangential at 0 and 45 degrees as expected, but has a
positive radial component for angles in between (and a negative
radial component for angles between 45 and 90 degrees.) In addition,
I found that the field increases monotonically in magnitude as one
moves from 0 to 45 degrees. These are exactly the signatures one
would expect for the squarish solution and I would further expect the
boundary conditions to require the "squarish solution" to propagate
into the inner region.

John Mallinckrodt
Cal Poly Pomona
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Forum for Physics Educators
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