Harbor Loop

This was just a "trick question?" the professor told us on friday. Its not homework but he said "think about it on the weekend"
and I've been thinking about it!

A `harbor loop' is a coil of wire of extensive area laid
out on the floor of a harbor entrance to detect the en-
trance of submarines by means of the voltage induced
by the submarine's natural magnetism. Suppose the
submarine is prolate in shape, and the submarine's
magnetic field happens to be the same as that of a
giant toroid, with the axis of the toroid along the axis
of the submarine and the radius of the toroid much
larger than the radius of the submarine. Further sup-
pose that the radius of the toroid is larger than the
depth of water above the harbor loop. If the harbor
loop is rectangular, is it possible for the submarine
to enter the harbor undetected? If so, under what
conditions? If not, why not?
Here is my drawing:
http://img502.imageshack.us/img502/4705/harborloopjpg.jpg [Broken]
http://img502.imageshack.us/img502/4705/harborloopjpg.jpg [Broken]

A toroid keeps its magnetic field inside the coils, therefore in the middle of the toroid there wouldn’t be a magnetic field. If the toroid’s radius is larger than the dept of the harbor loop and the submarine is submerged, the loop would be inside the torid’s inner radius in which there is no magnetic field. The submarine would pass undetected by the loop. The submarine can be seen if its floating instead of submerged (a person could see it).

And this is what i was thinking, I emailed him but he told me "Your answer involves a rather extreme assumption -- one extreme enough to ensure that your answer is not a complete answer."

then I was thinking and tirpping out and came up with random stuff:

Taking into consideration the "width" of the loop, maybe assuming that its much larger (or at least as large) as the toroid's radius. Assuming the toroid's field does pass trough the harbor loop

If the magnetic flux is zero, there is no induced current. The magnetic flux is given by the dot product of B with dA, if they are perfectly perpendicular then theta would be π/2 therefore giving a zero flux. The submarine can't pass undetected due to the shape of the rectangular loop. If the toroid is perfectly aligned with the "horizontal" components of the loop at a right angle, the flux would be zero (since it's a dot product and the scalar version has a cosθ) Only the component parallel to the field would be detected. This can be avoided if the toroid's field is aligned with the earth's magnetic field and is the same magnitude as the earths magnetic field. The current is induced by a change of magnetic flux, if the fields are the same then there is no change in flux, the horizontal components would cancel due to the dot product and the other components would cancel if it were aligned to the earth's magnetic field.

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Based on your description of the alignment of the toroidal B field and the harbor loop, your analysis is correct. Such a toroidal B field can be generated only by axial currents in the submarine. The actual B field is a dipole field, depending on the Earth's magnetic field at the shipyard where the submarine was built. Here is a description of the actual harbor loops used from about 1914 to 1950, and the magnetic field in the submarines.
http://indicatorloops.com/loopworks.htm
Nowdays, the very complicated system of rotating mirrors and photocells in the electromechanical loop integrator circuit could be replaced with several RC integrator operational amplifers.
Bob S

well that picture is MY interpretation of the problem.
as I mentioned I sent that to my professor and he told me
"nice try, but you are making an extreme assumption, so extreme that your answer is incomplete!"

and i dont know WTF he is talking about, he wont give me anymore hints!

The illustrations in
http://indicatorloops.com/loopworks.htm
show that the magnetic field direction in a submarine is down, but has a longitudinal component (along the axis of the submarine).. Only the longitudinal component will induce a voltage in the harbor loop. For the double loop in the illustrations, the voltage pulse is tripolar as shown, with a zero net area.
Bob S

The illustrations in
http://indicatorloops.com/loopworks.htm
show that the magnetic field direction in a submarine is down, but has a longitudinal component (along the axis of the submarine).. Only the longitudinal component will induce a voltage in the harbor loop. For the double loop in the illustrations, the voltage pulse is tripolar as shown, with a zero net area.
Bob S

well yeah, but the problem states that the submarine has the magnetic field different from conventional submarines, it says that it has the same field as a giant torroid.

I believe my answer is correct but as the professor said "incomplete"
he told me I made an extreme assumption which honestly i dont know what it is!

The theoretical inductance of the loop changes, but I don't see how that will affect the Faraday induction at low frequencies. But the mere presence of the submarine, even if it was unmagnetized, does affect the direction of the Earth's magnetic field lines. This may preferentially produce a dipole moment along the axis of a prolate spheroid.
Bob S

The theoretical inductance of the loop changes, but I don't see how that will affect the Faraday induction at low frequencies. But the mere presence of the submarine, even if it was unmagnetized, does affect the direction of the Earth's magnetic field lines. This may preferentially produce a dipole moment along the axis of a prolate spheroid.
Bob S
that seems pretty interesting! I think you hit the jackpot! could you elaborate on that?

that seems pretty interesting! I think you hit the jackpot! could you elaborate on that?
I know that if a soft iron rod is placed in a magnetic field, it will align along the field, in either direction. (diamagnetic rods align at 90 degrees). I have seen this torque derived somewhere for paramagnetic prolate spheroids. The presence of torque relates to an induced axial magnetic field.
[added] This paper may be of use
http://www.physics.princeton.edu/~mcdonald/examples/EM/jackson_ajp_74_272_06.pdf [Broken]
Strong vertical magnetic fields make prolate spheroids like paperclips (and submarines?) stand on end. Weak fields just induce axial dipole moments.
Bob S

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