Exploring the Relationship Between Magnetic Fields and Current Density

In summary, the final equation states that the curl of the magnetic field at any point is proportional to the current density at that point.
  • #1
ObsessiveMathsFreak
406
8
the final equation

∇XB(x) = μ0j(x)


But this means that the curl of the magnetic field at any point is proportional to the current density at that point.

But take the case of a long straight wire carrying current.

The magnetic field surrounding the wire is circular and hence its curl is everywhere constant in value.

But that means that the current density is everywhere constant in value, even at a million miles away from the wire.

what's with that?
 
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  • #2
yeah, a million miles from the wire the current density is constant, it's zero. Anywhere outside the wire, the current density is zero.

JMD
 
  • #3
Originally posted by ObsessiveMathsFreak
the final equation

∇XB(x) = μ0j(x)


But this means that the curl of the magnetic field at any point is proportional to the current density at that point.

But take the case of a long straight wire carrying current.

The magnetic field surrounding the wire is circular and hence its curl is everywhere constant in value.

Outside the wire the curl vanishes since outside the wire j(x) = 0.

Pete
 
  • #4
But the curl doesn't vanish. outside the wire the magnetic field is circular, meaning it has a constant curl.
 
  • #5
Curl is zero where there is no current, pmb is correct. Curl and integral over extended loop are different quantities. Whan you integrate over loop you have to include sources (currents) if the loop includes them.
 
  • #6
Try computing the curl of a circular field somewhere other than the axis.
 
  • #7


But this means that the curl of the magnetic field at any point is proportional to the current density at that point.

I think I see the problem now. The magnetic field is *not* proportional to current density - the *curl* of the magnetic field is. Sorry I din't note that earlier.

Pmb
 
  • #8
Man, you guys call the second derivative "curl"?

blegch
 
  • #9
Originally posted by KillaMarcilla
Man, you guys call the second derivative "curl"?

blegch

No, the curl is the differential operator:

[nab]×

which acts on vector fields. It is not the second derivative.
 
  • #10
Oh, right, I think I know what you're talking about now

Sorry, I had Math 126 about two years ago, and haven't used most of it since then (except for the geometric series approximations)

h0 h0, I look like quite the f00l now
 

1. What are Maxwell's Equations?

Maxwell's Equations are a set of four equations that describe the fundamental principles of electromagnetism. They were developed by physicist James Clerk Maxwell in the 19th century and are considered one of the most important and influential equations in physics.

2. What is the purpose of Maxwell's Equations?

The purpose of Maxwell's Equations is to describe the relationship between electric and magnetic fields, and how they interact with each other and with charged particles. They also provide a framework for understanding and predicting electromagnetic phenomena, such as light, radio waves, and electricity.

3. What are the four equations in Maxwell's Equations?

The four equations in Maxwell's Equations are Gauss's Law, Gauss's Law for Magnetism, Faraday's Law of Induction, and Ampere's Law with Maxwell's correction. These equations describe the behavior of electric and magnetic fields in relation to each other and to charged particles.

4. What is the significance of Maxwell's Equations in modern science and technology?

Maxwell's Equations have been foundational in the development of modern technology, particularly in the fields of electronics, telecommunications, and electromechanics. They have also been crucial in the understanding of electromagnetic waves and their use in fields like astronomy, meteorology, and medical imaging.

5. Are there any current research or applications related to Maxwell's Equations?

Yes, there is ongoing research and applications related to Maxwell's Equations, particularly in the field of electromagnetics. Scientists are constantly exploring new ways to apply these equations in fields such as wireless power transfer, metamaterials, and quantum computing. Additionally, there is ongoing research to further refine and expand upon Maxwell's Equations to better understand the behavior of electromagnetic fields in extreme conditions, such as in black holes or the early universe.

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