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I understand that a change in flux linkages of an inductor results in induced emf across it, the magnitude and direction of which can be determined by Faraday's and Lenz's laws. The magnitude of this emf can be given in terms of rate of change of current and inductance ($L \frac{di}{dt}$) as well as rate of change of flux linkages ($N \frac{d \phi}{dt}$). By equating the two, it appears that the value of Inductance is directly proportional to infinitesimal change in flux and inversely proportional to infinitesimal change in current. Does it mean that for a given change in current, if the value of Inductance is large, it results in a larger change in flux linkages? But doesn't it also mean that a larger inductance allows a larger flux change, which sounds contradictory to the original definition that a large inductance will oppose the change in flux more? I am not able to understand where I am making a mistake in understanding?

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You're right on both counts. For a given $\frac{di}{dt}$, big $L$ means big $\frac{d\phi}{dt}$ (your first statement) since $\frac{d\phi}{dt}= \frac{L}{N}\frac{di}{dt}$. At the same time, the "opposition to the change in flux" is a flux (not a change of flux), produced by a current, produced by the back-emf, and the back-emf is $L\frac{di}{dt}$. In other words, if $L$ is big, a given $\frac{di}{dt}$ induces a big voltage, and hence a big $i$, which produces a big $\phi$ in opposition to $\frac{d\phi}{dt}$.

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