Calculating Magnetic Field Lines

In summary, it is possible to calculate the field lines of a magnetic field using the equations for the magnetic field.
  • #1
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I guess this is maybe more of a vector calculus question, but here it goes. Say I have an arbitrary current distribution [itex]\vec{J}[/itex] with the corresponding magnetic field given by
[tex]\vec{\nabla}\times\vec{H} = \vec{J}[/tex]
[tex]\vec{\nabla}\cdot\vec{H} = 0[/tex]
What is the best way to calculate the field/flow lines of [itex]\vec{H}[/itex] (contours to which [itex]\vec{H}[/itex] is always tangent)? Do I need to calculate the magnetic field and then find the field lines from that, or is there an easier way to directly extract information about the geometry of the field lines from the equations above if I don't care about the magnitude of the magnetic field?

Edit: If it makes things easier, I'd still be interested to see the answer to this question with the added assumption that everything is uniform in the z-direction and [itex]\vec{J}[/itex] only has a z component (i.e. the magnetic field is 2D: it has no z-dependence and no z-component)
 
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  • #2
If you want H in a source-free region (no currents or magnetization), then Laplace's equation holds and you can define an effective magnetic scalar potential that is analogous to the usual electric potential. Equipotential lines and streamlines then play the same role as in electrostatics problems: you get orthogonal potential lines and field lines. This is by far the easiest approach.

Since J has only a z component, the problem is two-dimensional problem, further simplifying it.
 
  • #3
marcusl said:
If you want H in a source-free region (no currents or magnetization), then Laplace's equation holds and you can define an effective magnetic scalar potential that is analogous to the usual electric potential. Equipotential lines and streamlines then play the same role as in electrostatics problems: you get orthogonal potential lines and field lines. This is by far the easiest approach.

Since J has only a z component, the problem is two-dimensional problem, further simplifying it.

That does make sense, but unfortunately I do care about being able to do it inside the source region as well. Is there any simple technique there?
 
  • #4
No, nothing simple. You typically work with the vector potential. Complexity will depend on the source geometry and boundary conditions.
 
  • #5
thegreenlaser said:
I guess this is maybe more of a vector calculus question, but here it goes. Say I have an arbitrary current distribution [itex]\vec{J}[/itex] with the corresponding magnetic field given by
[tex]\vec{\nabla}\times\vec{H} = \vec{J}[/tex]
[tex]\vec{\nabla}\cdot\vec{H} = 0[/tex]
What is the best way to calculate the field/flow lines of [itex]\vec{H}[/itex] (contours to which [itex]\vec{H}[/itex] is always tangent)? Do I need to calculate the magnetic field and then find the field lines from that, or is there an easier way to directly extract information about the geometry of the field lines from the equations above if I don't care about the magnitude of the magnetic field?

The way I know how to do this requires you to calculate the field first. Once you know [itex]\mathbf{H}(\mathbf{r})[/itex]then you just construct the equations for the field lines. Let [itex]d\mathbf{r}[/itex] be a differential vector parallel to the magnetic field; then we know [itex]d\mathbf{r} \times \mathbf{H} = 0[/itex]. This yields differntial equations for the field lines. For example, in Cartesian coordinates this yields
[tex]
\frac{dx}{H_x} = \frac{dy}{H_y} =\frac{dz}{H_z}
[/tex]
and in spherical coordinates this is
[tex]
\frac{dr}{H_r} = \frac{rd\theta}{H_\theta} =\frac{r \sin\theta d\phi}{H_\phi}
[/tex]
jason
 
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What is a magnetic field line?

A magnetic field line is an imaginary line that represents the direction and strength of a magnetic field at a particular point in space. It is often used to visualize the shape and orientation of a magnetic field.

How do you calculate magnetic field lines?

Magnetic field lines can be calculated using the formula B = µ0I/2πr, where B is the magnetic field strength, µ0 is the permeability of free space, I is the current, and r is the distance from the source of the field.

What is the importance of calculating magnetic field lines?

Calculating magnetic field lines is important in understanding the behavior of magnetic fields and their effects on charged particles. It is also crucial in various applications such as in the design of magnetic devices and in the study of natural phenomena like Earth's magnetic field.

What factors affect the shape of magnetic field lines?

The shape of magnetic field lines is affected by the strength and direction of the magnetic field, as well as the presence of other magnetic fields nearby. The shape can also be influenced by the type of material the field is passing through.

Can magnetic field lines intersect?

No, magnetic field lines cannot intersect because it would imply that the magnetic field has two different directions and strengths at the same point in space, which is not possible. They can only touch at a single point or form closed loops.

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