Exploring the Magnetic Field: Applying Maxwell's Equation and Stokes' Theorem

In summary, the conversation discusses Maxwell's equation about the divergence of a magnetic field being zero, which leads to the existence of a vector function and the application of Stoke's theorem. However, there is a question about applying Stoke's theorem in this case due to the region not being simply connected.
  • #1
typhoonss821
14
1
One of Maxwell's equations says that [tex]
\nabla\cdot\vec{B}{=0}
[/tex] where B is any magnetic field.
Then using the divergence theore, we find
[tex]
\int\int_S \vec{B}\cdot\hat{n}dS=\int\int\int_V \nabla\cdot\vec{B}dV=0
[/tex].

Because B has zero divergence, there must exist a vector function, say A, such that
[tex]
\vec{B}=\nabla\times\vec{A}
[/tex] .
Combining these two equations, we get
[tex]
\int\int_S \hat{n}\cdot\nabla\times\vec{A}dS=0
[/tex] .
Next we apply Stoke's theorem and the preceding result to find
[tex]
\oint_C\vec{A}\cdot\hat{t}ds=\int\int_S\hat{n}\cdot\nabla\times\vec{A}dS=0
[/tex] .
Thus A is path independent. It follows that we can write
[tex]
\vec{A}=\nabla\psi
[/tex] , where ψ is some scalar fution.

Since the curl of the gradient of a function is zero, we arrive at the remarkable fact that
[tex]
\vec{B}=\nabla\times\nabla\psi=0
[/tex]
that is, all magnetic field is zero!

Wow, there must be something wrong...

My thought is that we can not apply Stoke's theorem in this case because the region we discuss is not simply connected.

But I'm not sure if I am right, please help me check the proof^^
 
Last edited:
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  • #2
The surface you use for the divergence theorem is a closed surface that encloses a volume. The surface is from the Stokes theorem is a surface that has the contour C as edge.
 

1. What is a magnetic field and how is it created?

A magnetic field is a region in space where a magnetic force can be detected. It is created by moving electric charges, such as electrons, which generate a magnetic field around them. This can also be seen in the behavior of magnets, which have their own magnetic fields.

2. How do Maxwell's equations and Stokes' theorem relate to exploring the magnetic field?

Maxwell's equations are a set of fundamental equations that describe the behavior of electric and magnetic fields. Stokes' theorem is a mathematical tool used to calculate the circulation of a vector field, such as a magnetic field. By applying these equations and theorem, scientists can better understand and predict the behavior of magnetic fields.

3. What are some real-world applications of exploring the magnetic field?

Exploring the magnetic field has many practical applications, including in technology such as motors, generators, and MRI machines. It is also critical for understanding the Earth's magnetic field and its role in navigation and protecting us from harmful solar radiation.

4. Can a magnetic field be manipulated or controlled?

Yes, a magnetic field can be manipulated and controlled using various methods, such as changing the direction of electric current or using electromagnets. This allows for the design and development of various technologies that rely on magnetic fields.

5. How do scientists measure and map the magnetic field?

Scientists use instruments such as magnetometers and gaussmeters to measure the strength and direction of a magnetic field. They can also create maps of magnetic fields by taking measurements at different points and using mathematical models to interpolate the data and visualize the field's shape and strength.

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