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xman
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I am trying to calculate the current induced in a loop assuming a magnetic monopole exists. The loop is a perfect conductor, which I understand implies the electric field inside must be zero. I picture the problem with the magnetic monopole traveling with a velocity coaxial with the loop. I am given the loop has a self inductance L, so what I've done is take Faraday's law (modified for the existence of a mag. monopole), and the current density I'm associating as follows
[tex] \vec{\nabla} \wedge \vec{E} =0= -\left\{ \mu_{0} \vec{j}_{m}+\frac{\partial \vec{B}}{\partial t} \right\}[/tex]
Yielding
[tex] \mu_{0} \vec{j}_{m} = -\frac{\partial \vec{B} }{\partial t} [/tex]
Now relating to the induced emf I have
[tex] -L\frac{dI}{dt} = -\dot{\phi}_{m} = \int -\frac{\partial \vec{B}}{\partial t} \cdot d\vec{a} [/tex]
From which I make the identification
[tex] \vec{j}_{m} = \rho_{m} \vec{v} [/tex]
Which gives assuming a planar area loop with the direction of the velocity being parallel to the loop is given by
[tex] -L \frac{dI}{dt} = \mu_{0} \rho_{m} A_{loop} v
\Rightarrow -L\frac{dI}{dt} =\mu_{0} \rho_{m} A_{loop} \frac{dz}{dt} [/tex]
which is where I am stuck. I think I want to integrate over a long time, including the point and past the point the magnetic monopole passes through the loop. My question(s) is does everything I have done so far make sense, and if so where do I go from here?
[tex] \vec{\nabla} \wedge \vec{E} =0= -\left\{ \mu_{0} \vec{j}_{m}+\frac{\partial \vec{B}}{\partial t} \right\}[/tex]
Yielding
[tex] \mu_{0} \vec{j}_{m} = -\frac{\partial \vec{B} }{\partial t} [/tex]
Now relating to the induced emf I have
[tex] -L\frac{dI}{dt} = -\dot{\phi}_{m} = \int -\frac{\partial \vec{B}}{\partial t} \cdot d\vec{a} [/tex]
From which I make the identification
[tex] \vec{j}_{m} = \rho_{m} \vec{v} [/tex]
Which gives assuming a planar area loop with the direction of the velocity being parallel to the loop is given by
[tex] -L \frac{dI}{dt} = \mu_{0} \rho_{m} A_{loop} v
\Rightarrow -L\frac{dI}{dt} =\mu_{0} \rho_{m} A_{loop} \frac{dz}{dt} [/tex]
which is where I am stuck. I think I want to integrate over a long time, including the point and past the point the magnetic monopole passes through the loop. My question(s) is does everything I have done so far make sense, and if so where do I go from here?
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