Given a direction of current in a wire, and the magnetic field perpendicular to the wire forms with say "north" pointing clockwise (right-hand-rule convention). If we placed a small monopole object corresponding to "south" in the field around the wire, would it form a clockwise helix around the wire since it's only attracted to the "north" of the field, and continuously loop around the wire until other forces take it off the path?
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3$\begingroup$ You are free to solve Maxwell's equations in the presence of a magnetic charge density, if you so choose. You just write $\vec \nabla \cdot \vec B = 4\pi \rho_M$ instead of $\vec \nabla \cdot \vec B = 0$. (See Schwinger's textbook on Electrodynamics for a treatment that carries around the magnetic charge density for no reason.) $\endgroup$– hftCommented Apr 24 at 19:56
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$\begingroup$ Hi hft, for my clarity, the [4πρM] instead of 0 allows us specifically to look at a monopole which is simply a magnetic charge, am I following correctly? $\endgroup$– TeragregCommented Apr 24 at 22:22
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1$\begingroup$ It allows you to look at any magnetic charge distribution. The $4\pi$ is just because I am used to Gaussian units: en.wikipedia.org/wiki/Gaussian_units#Maxwell's_equations $\endgroup$– hftCommented Apr 24 at 23:08
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$\begingroup$ Here is a similar question: physics.stackexchange.com/questions/129401/… $\endgroup$– hftCommented Apr 24 at 23:09
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1$\begingroup$ That’s a very good question. Thanks for your help. Follow-up to the first question. I have ordered the QED book per your reference as this is an area of physics that never ceases to amaze, and the book is a great compilation. $\endgroup$– TeragregCommented Apr 24 at 23:13
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Hint: you might solve the problem in plane polar coordinates, since your monopole will never leave the plane perpendicular to the current-carrying wire. The magnetic field induced by the wire will vary as b/r, so the equations of motion will be $$ \ddot{\vec r}=\hat{r} (\ddot r -r\dot\theta^2)+ \hat\theta (r\ddot\theta+2\dot r \dot \theta)= \hat \theta {b\over r} $$ so that $$ \ddot r = r\dot\theta ^2, ~~~~ r\ddot\theta+2\dot r \dot \theta={b\over r}~~. $$ You may solve these and perhaps eliminate t from them, so as to express your parametric spiral, $r(\theta)$ .
answered Apr 24 at 19:07
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$\begingroup$ Hi Cosmas, thank you for response. If I'm following you, [r¨=rθ˙2] is the change in radial distance across time, and [rθ¨+2r˙θ˙=b/r] is the change in angle? I was only asking if the monopole in such a situation would behave by circling the wire, never mind describing its path! But your response is a far more detailed and higher level response than I anticipated and i appreciate it! $\endgroup$– TeragregCommented Apr 24 at 22:04
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1$\begingroup$ Yes, the second equation monitors the change in the polar angle. It determines the spiral, whose precise form I have not recognized, myself. $\endgroup$ Commented Apr 24 at 22:24