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In this source it employs KVL: http://physics.info/circuits-rlc/

BUT in this source at page 6 it says "Apply Faraday’s Law in order to find I(t). (Can’t use KVL since magnetic flux is changing)": http://aplusphysics.com/courses/ap-c/tutorials/APC-Electromagnetism.pdf

Many texts use KVL.What's going on?

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It appears to me that they are in fact using Kirchoff's Voltage Law, using the differential relation between current and voltage for an inductor with self-inductance, L:

$$V=-L\frac{\mathrm{d}I}{\mathrm{d}t}$$

Note that this is in fact derived from Faraday's Law in this case, cf. the line:

$$\oint \vec{E}\cdot\mathrm{d}\vec{\ell}=-\frac{\mathrm{d}\Phi}{\mathrm{d}t}=-L\frac{\mathrm{d}I}{\mathrm{d}t}$$

Where the LHS of the equation is the closed loop voltage and the RHS is the familiar differential relation from the center of the equation.

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