Hi,
According to the Biot-Savart law, the magnetic field of an infinite wire dies off proportional to 1/r, where r is the distance to the wire.
Please, could anyone point me our if there exits any other geometrical configurations that have magnetic field dies off proportional to 1/r?
Thank you.
Wang Zhe
in principle, for any filiform circuit (C)traversed by a current dI, the resulting magnetic field is proportional to 1 / r. The Biot et Savart law is clearly expressed as :
dB =m0/4pi I dlxr/r3 (in the MKSA system)
where I the electric current
dl : element of circuit C in the same sens of the current I
By a simple dimensional analyze, you can see that B will be in 1/r in these conditions (wire, solenoid,..)
In the static case, perhaps one of the four Maxwell equation (Ampere's circuital law) is even simpler to apply: In the absence of any displacement current, the integral of B along a closed loop is proportional to the integral of the current through the area enclosed by the loop. Consequently, if the current is restricted to a certain area, the integral of B along a circle enclosing this area is independent of the radius r of the circle. Since the circumference is proportional to r, if B is constant along the circle, it has to be proportional to 1/r.
But the case of the straight infinite wire is special because it is rotationally symmetrical; so B is really independent of the angle. It is different in all other configurations I can think of, perhaps the simplest one the solenoid of infinite length: There is a homogeneous B field inside, and no B field at all outside.
If your question is not limited to the static case: If the distance to the source of omnidirectional electromagnetic radiation is large enough to treat the source as a point, the power density S of the radiation has to be proportional to 1/r2 due to conservation of energy. Since S = E x H, and the ratio E/H remains constant, both the E field and the H field (and hence B) have to decrease proportionally to 1/r, provided the medium in which the radiation propagates is homogeneous.
Dear Prof Hafidi and Prof Fricke,
Thank you so much for your replies!
If we consider placing a magnetic dipole (say a small current loop) inside an infinitely long cylinder. Assuming the radius of the loop is of order of the cylinder, then the system is essentially dimensional.
I know that the far-field magnetic field due to a dipole falls off like 1/r^3. However, someone pointed out that the magnetic dipole in this particular case (i.e., infinitely long cylinder) dies off like 1/r^2.
Please, could you confirm me?
Thank you.
Wang Zhe
Dear Zhe Wang,
is the cylinder filled with a material or is it a cylindrical shell? What are the magnetical properties of the material?
Dear Prof Fricke,
I consider the induced magnetic field vanishes at the boundary of the cylinder like
B(r)=B_c (1-r^2/R^2),
where B_c is the induced magnetic field at the center line of the cylinder, r is the distance away from the center, and R is the radius of the cylinder.
Thank you.
Wang Zhe
Dear Prof Fricke,
In fact, I use the electromagnetic analogy to investigate the fluid dynamics problem. In fluid dynamics, the velocity induced by a vortex filament obeys a relation resembling the Biot-Savart law
dB =Gamma/4pi I dlxr/r3,
where Gamma is the strength (circulation) of the vortex filament.
Let us consider the flow in a circular pipe, where the mean flow profile is parabolic (velocity vanishes at the pipe wall). So that I suppose that the cylinder in the previous post should be considered as an insulator?
My aim is to get the analytic form of kinetic energy of the velocity field due to the vortex filament, which is complicated to calculate. Thus, instead, I try to figure out the far-field distribution of velocity, i.e., 1/r, 1/r^2, and 1/r^3.
Assuming the boundaries of the system are perfect insulators, I would also like to know that if the geometry of the system will change qualitatively the velocity distribution (e.g., 1/r^3 to 1/r^2).
Thank you!
Wang Zhe
Dear Zhe Wand, while J. C. Maxwell used the geometrical analogy between incompressible fluids and electric and magnetic force fields as a step towards his equations, and while some designations such as "magnetic flux" are reminiscent of this analogy, I doubt that one can apply this analogy more generally.
One reason is that fluids can be truly confined to an arbitrary volume of space while fields cannot be "shielded". That is, any effective shielding boils down to generating additional fields which, together with the primary field, sum up to zero in a certain spatial region. So, while the flow of a fluid inside a pipe and the magnetic field inside a coil (or a core) might look similar, the underlying mechanism is quite different. In other words, there is no "magnetic insulator" analogical to a pipe.
Another reason is the difference regarding energy conversion: A static magnetic field (generated, for example, by a permanent magnet) is energy resting in space. The analogical fluid distribution would be a mixture of potential (pressure) and kinetic (velocity) energy; generally, part of the kinetic energy would be converted into thermal energy. This conversion might generate vortices. There is nothing analogical to this generation of vortices in the case of the magnetic field: A magnetic vortex is accompanied either by a flow of charges and/or by a temporally changing electric field.
Although my remarks might sound discouraging, I hope that they are helpful.
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