If you take a bunch of random particles and put them together, why should a pole form on each side of this collection?
Some particles already have a magnetic field. Many particles are polar, such that they will orient themselves in a magnetic field. If you jumble them all together, they will self align, and eventually one strong field will be externally detectable even though their individual fields were small and unorganized at the start.
Perform this experiment: Drop a bunch of magnetic powder and dirt into a bag. Shake vigorously. What is the resulting clump's magnetic signature?
Is it in practice possible to create a device capable of canceling the earth's magnetic field in a region the size of the north sea?
No. What you want is a Helmholtz coil, adjusted electronically to react to the earth's changing field.
However, the area of the field required, even though it would be relatively low magnetic force, would require entirely too much energy to be practical. Further, an ideal Helmholtz coil, where the field is uniformly 0 everywhere inside the coils, requires essentially a cubic structure. The North Sea is 970 KM long, and thus the coils would need to be 970KM in diameter, vertically oriented, buried a significant portion of that depth into the ground on either side of the north sea.
Further, it would really mess up the compasses of people traveling anywhere near the coils, not to mention other animals that appear to depend on magnetic fields, such as some migrating birds.
This is super belated, but I'll try to give an answer to this question, should it be helpful for future homework doers. As is, the question is a bit under-specified because it may not be evident what frequency is being mentioned (resonance, precession, etc., though essentially the same number), which is probably why answers have not been forthcoming.
The short answer is YES, your approach is correct, but I'll make a reasonable assumption of precessional motion being detected, and explain why, based on the assumptions. In this experiment, one uses the Helmholz coil to do the cancellation of Earth's field (plus any other sources), but the magnet is used to provide detection. Basically, a magnet at some small angle with a total net magnetic field $B_T$ will undergo what is known as 'precession' (wobble) about the total magnetic field at a frequency $\omega_p$ proportional to
$\omega_p = 2\pi f \propto \gamma * B_T$
$\gamma$ is called the gyromagnetic ratio and is proportional to charge over mass. $B_T$ is the net field including the Earth's magnetic field, any stray field sources from magnets, and the field produced by the coil current. Your equation above is a form of $(1/2\pi)\gamma * B_T$, and/or the corresponding classical energy, proportional to $\omega^2$. So, when you cancel the earths magnetic field (plus any other stray magnetic field source), it will be observed when the precessional frequency vanishes or becomes zero, or the magnet's energy vanishes (ceases motion). Under these assumptions, your approach is correct :) Hopefully this explains a bit more as to why. So, well done, even though you are probably a Professor by now!
Best Answer
There is one and only way to cancel something: add its negative to itself. However, there is an alternative to cancellation for shielding a region from external electromagnetic fields.
Generally speaking, methods of isolating a region from external electromagnetic fields (EM shielding) can be divided into two categories, passive and active. A passive shield prevents the external field from reaching the isolated internal region. Whatever the field is outside, the field is zero inside. This is convenient if the strength of the external field is variable or unknown. Faraday cages (shields made from a mesh of conducting material) are examples of passive shields against static (and non-static) electric fields.
Alternatively, if you know the value of the external field from which you want to isolate a region, you can generate an equal and opposite field to the external field to actively "cancel it out". The active alternative to a Faraday cage for blocking electrostatic fields is a capacitor, whose geometry is precisely shaped so that the electric between the two charged plates exactly cancels the external field in the region of interest.
The magnetostatic analog for active shielding is field cancellation using solenoids with the appropriate geometry. The passive alternative for magnetostatically shielding a region (analog to the Faraday cage) is an enclosing surface made of material (metal alloys) with high magnetic permeability. They don't exactly block the external magnetic field per se in the same way a Faraday cage blocks an external electric field, but rather draw the field into themselves, providing a path for the magnetic field lines around shielded cavity.
However, the effectiveness of passive magnetostatic shielding diminishes for very weak fields. From a practical standpoint, this usually makes active shielding by electromagnets (solenoids, Helmholtz coils, etc.) the more useful of the two options.