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Your initial supposition, "If you have increasing current going down an inductor,..." is a bit vague about exactly what external influences are at work on the inductor in question. I will suppose that the current through the inductor is increasing because a battery has just been connected directly to the inductor. Other suppositions are possible, but this ...

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If you have a series LCR circuit then at every instant of time the current in each part of the circuit must be the same which is KCL - the conservation of charge. Let that current be $I=I_o \sin (\omega t)$ Now one must add up all the voltages in the circuit and make the sum zero. The complication is that although all the voltages will be of the same ...

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When you apply an AC source of given frequency to the components R, L and C in series the current through all the components must be the same. The use of vectors are introduced because for the different components the relationship between the voltage across the component and the current through them are different: For R the voltage is in phase with the ...

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