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VantagePoint72

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However, in many—most, even—circuits of interest it is a (possibly time dependent) voltage that's

*applied*across an inductor and current thus builds up according to the integral of the above equation. While I can see well enough how this works mathematically—the equation is valid whether ##I## or ##V## is the independent variable—I have a hard time turning the above argument around in order to understand physically what's happening. In terms of Maxwell's equation (and perhaps the Lorentz force law, if necessary) how can you derive the relation ##\frac{dI}{dt} = -V/L## when ##V## is applied across the inductor? To take the simplest example, consider an ideal inductor in series with a battery: the current through the inductor simply increases linearly as ##I(t) = \frac{V}{L}t## in the opposite direction as the applied voltage. How should I understand this behaviour from first principles?