What causes an electric field line to close on itself as opposed to other cases?
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1$\begingroup$ $\vec \nabla \cdot \vec E = 0$ $\endgroup$– ACuriousMind ♦Commented Dec 27, 2014 at 17:09
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$\begingroup$ Nothing. An electric "field line" is a line drawn by people on paper to visualize a vector field. $\endgroup$– CuriousOneCommented Dec 27, 2014 at 17:09
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2 Answers
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You are talking about a "divergence-free" field, otherwise known as a solenoidal field. In such cases, $\nabla \cdot \vec{E}=0$ and electric field lines have no beginning and no end.
From Maxwell's equations we see that $\nabla \cdot \vec{E}=0$ when the charge density is zero. So, for instance, electric field lines cannot begin or end in a vacuum or in an entirely neutral medium.
Magnetic fields are always divergence-free, because unlike electric fields, there are (probably) no monopoles that act as sources of B-field.
Divergence-free E-fields can be produced without free charges according to the Maxwell-Faraday law. $$ \nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$$
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Magnetic monopole (if they existed) currents and time-dependent magnetic inductions (from $\nabla\times\mathbf E = -\frac{\partial\mathbf B}{\partial t} + \mathbf J_m$).
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$\begingroup$ Magnetic monopoles would be the opposite case - magnetic fields that have a nonzero divergence and thus have field lines that do not form closed loops. All magnetic fields, as described by Maxwell's equations, have field lines that "close on themselves." $\endgroup$– user_35Commented Dec 27, 2014 at 17:14
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1$\begingroup$ my bad, i just confused monopoles with monopole current, as correctly indicated by the equation I wrote. $\endgroup$ Commented Dec 27, 2014 at 17:15
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