1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

I Determine the direction of magnetic field from Maxwell eqs?

  1. Sep 29, 2016 #1
    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?
     
  2. jcsd
  3. Sep 29, 2016 #2

    Simon Bridge

    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    2016 Award

    You have to solve the coupled differential equations for [1].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Determine the direction of magnetic field from Maxwell eqs?
Loading...