No. This fails due to your "arbitrarily closely". This cannot be true based on post #2. You cannot come arbitrarily close to a field with non-zero divergence using just divergence free fields.rumborak said:That said, I would think that in a limited space you should be able to approximate any field distribution arbitrarily closely, with enough surrounding magnets.
No you are not. The field will still have exactly zero divergence - just like the electric field of a point charge away from the point charge. As long as your definition of "close" is reasonable, you will always be able to find a field with non-zero divergence that is closer to the target field (with non-zero divergence) than any divergence free field and so you cannot get arbitrarily close.rumborak said:That depends on the definition of "close". True, of course you will never have a non-zero divergence, but if the incoming field lines are bundled very close together and then fan out radially, you are very close to emulating non-zero divergence.
In the case of Gibbs' phenomenon, you will not be able to find a function that better approximates the target function than all superpositions of sine waves because the sine waves form a dense basis and closeness here has a definite meaning in terms of the ##L^2## norm. This stands in stark contrast to the case of a divergence free field approximating a field with non-zero divergence.rumborak said:I am suggesting an "engineering approximation" here, where you try to confine the errors to an arbitrarily small region of space.
It's the same as with the Gibbs phenomenon when approximating a signal with sine waves. You will never get rid of the overshoot because it is a mathematical consequence, but if your definition of "close" is "minimize the amount of space with errors", you can get arbitrarily close.
The geometry of a magnetic field refers to the shape, direction, and strength of the magnetic field lines, which are invisible lines that show the force and direction of a magnetic field. These lines follow a specific pattern around a magnet or a current-carrying wire, forming closed loops that extend from the north pole to the south pole.
The geometry of a magnetic field is determined by the properties of the magnet or current-carrying wire, such as its shape, size, and strength. It also depends on the surrounding materials and the presence of other magnetic fields that may influence its shape and direction. In general, the geometry of a magnetic field is a result of the interactions between electric currents and charged particles.
The three main types of magnetic field geometries are dipole, quadrupole, and solenoidal. A dipole magnetic field has two poles, north and south, and its field lines extend from one pole to the other. A quadrupole magnetic field has four poles, with alternating north and south poles. A solenoidal magnetic field is a more complex geometry with multiple loops and twists, often found in electromagnets and in the Earth's magnetic field.
The geometry of a magnetic field can greatly affect its strength. In general, the closer the field lines are together, the stronger the magnetic field will be. This is why magnets with more tightly packed field lines are stronger. The shape and size of the magnet or current-carrying wire also play a role in determining the strength of the magnetic field.
Yes, the geometry of a magnetic field can be changed by altering the properties of the magnet or current-carrying wire, or by introducing other magnetic fields nearby. This can be done by changing the shape, size, or strength of the magnet or wire, or by manipulating the materials and currents around it. Electromagnets, for example, can change their geometry and strength by controlling the flow of electric currents through them.