Symmetry Arguments and the Infinite Wire with a Current

[Orodruin]

Many people reading this will be familiar with symmetry arguments related to the use of Gauss law. Finding the electric field around a spherically symmetric charge distribution or around an infinite wire carrying a charge per unit length are standard examples. This Insight explores similar arguments for the magnetic field around an infinite wire carrying a constant current $I$, which may not be as familiar. In particular, our focus is on the arguments that can be used to conclude that the magnetic field cannot have a component in the radial direction or in the direction of the wire itself.
Transformation properties of vectors
To use symmetry arguments we first need to establish how the magnetic field transforms under different spatial transformations. How it transforms under rotations and reflections will be of particular interest. The magnetic field is described by a vector $\vec B$ with both...

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[eljose79]

[/url]Hello, thank you for bringing up this interesting topic! I completely agree that symmetry arguments are extremely useful in understanding and solving problems in electromagnetism. In fact, these arguments are not limited to just the examples mentioned in the post, but can also be applied to more complex situations such as a current-carrying loop or a solenoid.

The key concept behind these symmetry arguments is the transformation properties of the electric and magnetic fields. As you mentioned, for a spherically symmetric charge distribution, the electric field can only have a radial component due to the symmetry of the distribution. Similarly, for an infinite wire carrying a constant current, the magnetic field can only have a tangential component due to the symmetry of the wire. This is because any radial or longitudinal components would violate the symmetry of the system.

Furthermore, these symmetry arguments are not limited to just the direction of the fields, but also the magnitude. For example, for a spherical charge distribution, the magnitude of the electric field decreases with distance from the center due to the inverse square law. Similarly, for an infinite wire carrying a constant current, the magnitude of the magnetic field decreases with distance from the wire due to the inverse distance law.

These arguments are not only useful in understanding the behavior of the fields, but also in solving problems. By using symmetry arguments, we can simplify complex situations and reduce them to simpler, more manageable cases. This not only saves time and effort, but also helps us gain a deeper understanding of the underlying principles.

In conclusion, symmetry arguments are a powerful tool in the study of electromagnetism and are essential in understanding and solving problems related to electric and magnetic fields. Thank you for sharing this Insight and bringing attention to this important topic.