I was trying to apply all 4 maxwell's equations to various real life situations and it gave some trouble. Suppose I had a battery(Ideal) and a loop of wire(conducting) and I used the battery to set up a steady current(assume battery maintains a constant EMF).now the curl of E=0 as long as it is a magnetostatic situation (dB/dt=0) from maxwells equations.But clearly the electric field has a non zero curl(non conservative) as it is directed along the loop even though in steady current. NOTE:I am assuming Drude's Model and concluding about drift velocity.
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asked Aug 10, 2023 at 13:59
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$\begingroup$ By definition, an ideal battery has zero internal resistance, so any resistor connected tot he battery will automatically see a completely closed current path through this ideal voltage source, by definition (again). The electric field along this network is as curl-free as can be. $\endgroup$– hyportnexCommented Aug 10, 2023 at 15:22
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$\begingroup$ Note that the battery will create it's own electric field. So, $\oint_C \mathbf E \cdot d \mathbf r = \varepsilon$. Where $\varepsilon$ is the electromotive force generated by the battery. $\endgroup$– Álvaro RodrigoCommented Aug 10, 2023 at 16:22
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You are correct, in a magnetostatic situation we have \begin{equation} \nabla \times \mathbf{E} = 0. \end{equation} But I think you are wrong that in the situation you described I think that the curl is still zero. Let's maybe look at a simple capacitor as that situation may be a little more instructive. So we have a capacitor which is charged at $t=0$. We also connect both of its end with a wire and put a load in between. Let's also assume that the flowing currents are small and the capacity as well as the charge of the capacitor are big so that we can approximate that the electric fields don't change over the time of observation. In the wire (and load) we have an electric field, which always shows into the direction of current flow BUT this loop isn't closed at the capacitor! In the contrary, the electric field between the plates in the capacitor shows in the opposite direction! Therefore the curl of the electric field is zero (as you could calculate, but I think my heuristic argument shows the point). The situation should be similar in a battery.
