Determine the direction of magnetic field from Maxwell eqs?

In summary, the conversation discusses the use of the right hand grip rule to determine the direction of a magnetic field induced by a current carrying wire, and whether this direction can also be deduced from Maxwell's Equations. The conversation then delves into the relevant equations and their interpretations, particularly focusing on the role of the phi component in the B Field. The speaker questions their previous interpretations and asks for clarification.
  • #1
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Hi! My high school physics tells me using right hand grip rule to determine the direction of magnetic field induced by a current carrying wire, but I wonder whether I can deduce the direction merely from Maxwell's Equations?

Suppose now we have a current density in cylindrical coordinates$$\vec{J}=J_{0}\hat{z}, \hspace{1 cm} 0 \leqslant \rho \leqslant r,$$

And the two relevant Maxwell's equations are:
$$\nabla \times \vec{B}=\mu_{0}\vec{J}\hspace{1 cm}[1]$$
$$\oint\limits_{\partial_\Sigma} \vec{B} \cdot d\vec{l}=\mu_{0}\iint\limits_{\Sigma}\vec{J}\cdot d\vec{S} \hspace{1 cm} [2]$$
Remarks: $$\frac{\partial \vec{E}}{\partial t}=0$$

From [1], I try to express the curl in cylindrical form:
$$(\frac{1}{\rho}\frac{\partial B_{z}}{\partial \phi}-\frac{\partial B_{\phi}}{\partial z})\hat{\rho}+(\frac{\partial B_{\rho}}{\partial z}-\frac{\partial B_{z}}{\partial \rho})\hat{\phi}+\frac{1}{\rho}(\frac{\partial (\rho B_{\phi})}{\partial \rho}-\frac{\partial B_{\rho}}{\partial \phi})\hat{z}=\mu_{0}J_{0}\hat{z}$$

How can I deduce the B Field only has a phi component from the above expression?

From [2], I try to interpret in this way:
suppose I set up a circular Amperian loop on the rho-phi plane, only the phi component of B Field contributes to the path integral on the left hand side, and the right hand size is the current, so I conclude the current induces B-Field with phi component only. But I hesitate immediately, what if the loop I set is inclined? I will immediately conclude the B-Field only has the inclined component using the previous argument, but this is not true! I must think something wrong. What do you think?
 
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  • #2
You have to solve the coupled differential equations for [1].
 

Related to Determine the direction of magnetic field from Maxwell eqs?

1. What are Maxwell's equations?

Maxwell's equations are a set of four equations that describe the relationship between electric and magnetic fields, and how they are affected by charges and currents in a given space. They are fundamental to the study of electromagnetism and have been instrumental in the development of modern technology.

2. How do Maxwell's equations help determine the direction of a magnetic field?

One of Maxwell's equations, known as the Ampere's law, relates the magnetic field to the current flowing through a given surface. By using this equation along with other equations, such as the Biot-Savart law, the direction of a magnetic field can be determined.

3. Can Maxwell's equations be applied to all types of magnetic fields?

Yes, Maxwell's equations can be applied to any type of magnetic field, including those found in everyday objects and those found in more complex systems, such as in the Earth's magnetosphere or in particle accelerators.

4. Are there any limitations to using Maxwell's equations to determine the direction of a magnetic field?

While Maxwell's equations are incredibly powerful and versatile, there are some limitations to their use. For example, they may not accurately describe the behavior of magnetic fields in extreme conditions, such as near black holes or in highly energetic environments.

5. How are Maxwell's equations used in practical applications?

Maxwell's equations have numerous practical applications, including in the design of electrical and electronic devices, in the study of the Earth's magnetic field, and in various medical imaging techniques. They are also essential in the development of new technologies, such as wireless communication and renewable energy sources.

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