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In volume two of the Feynman Lectures on Physics, Chapter 18, here: http://www.feynmanlectures.caltech.edu/II_18.html

There is a scenario in which an infinitely extending charged sheet is suddenly kicked into motion, parallel to its orientation. Electrostatic effects are being ignored here. Once this happens, a current is set up, and a 'wave' of magnetic field is propagated away from the sheet at a velocity $v$ in both directions, as shown in the following diagrams (taken from the Feynman Lectures website):

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I understand the indicated directions of the magnetic field lines, this is a standard sheet current setup. Why are the induced electric field lines pointing in the indicated direction? Is this obvious from Faraday's Law, $\nabla{\times}{\textbf{E}} = -\frac{\partial{\textbf{B}}}{\partial{t}}$? Should the direction of the electric field lines not change on either side of the sheet?

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By what I can understand from the picture, this is a simple application on Faraday's Law and Lenz Law. Yes, Faraday's law can be used to show that since there is a change in the magnetic flux that passes through the rectangle, there will be an induced emf. Nevertheless,to find the direction of this induced emf you should apply Lenz Law, which basically says that the induced emf will produce a magnetic field that will oppose the change that caused it. In this case the magnetic field points into the page and since the area of rectangle in this magnetic field is decreasing, the net magnetic flux will also decrease. This causes the induced emf to create a magnetic field that opposes this decrease in the magnetic field that points into the page, which means that a magnetic field pointing out of the page is created. By using the RH rule you can find the direction of current.

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