Summarizing Inductance

Summary

  • Inductance is the property of a device that tells how effectively it induces an emf in another device.
  • Mutual inductance is the effect of two devices in inducing emfs in each other.
  • A change in current \(\Delta {I}_{1}/\Delta t\) in one induces an emf \({\text{emf}}_{2}\) in the second:

    \({\text{emf}}_{2}=-M\cfrac{\Delta {I}_{1}}{\Delta t}\text{,}\)

    where \(M\) is defined to be the mutual inductance between the two devices, and the minus sign is due to Lenz’s law.

  • Symmetrically, a change in current \(\Delta {I}_{2}/\Delta t\) through the second device induces an emf \({\text{emf}}_{1}\) in the first:

    \({\text{emf}}_{1}=-M\cfrac{\Delta {I}_{2}}{\Delta t}\text{,}\)

    where \(M\) is the same mutual inductance as in the reverse process.

  • Current changes in a device induce an emf in the device itself.
  • Self-inductance is the effect of the device inducing emf in itself.
  • The device is called an inductor, and the emf induced in it by a change in current through it is

    \(\text{emf}=-L\cfrac{\Delta I}{\Delta t}\text{,}\)

    where \(L\) is the self-inductance of the inductor, and \(\Delta I/\Delta t\) is the rate of change of current through it. The minus sign indicates that emf opposes the change in current, as required by Lenz’s law.

  • The unit of self- and mutual inductance is the henry (H), where \(1 H=1 \Omega \cdot \text{s}\).
  • The self-inductance \(L\) of an inductor is proportional to how much flux changes with current. For an \(N\)-turn inductor,

    \(L=N\cfrac{\Delta \Phi }{\Delta I}\text{.}\)

  • The self-inductance of a solenoid is

    \(L=\cfrac{{\mu }_{0}{N}^{2}A}{\ell }\text{(solenoid),}\)

    where \(N\) is its number of turns in the solenoid, \(A\) is its cross-sectional area, \(\ell \) is its length, and \({\text{μ}}_{0}=4\pi ×{\text{10}}^{\text{−7}}\phantom{\rule{0.25em}{0ex}}\text{T}\cdot \text{m/A}\phantom{\rule{0.10em}{0ex}}\) is the permeability of free space.

  • The energy stored in an inductor \({E}_{\text{ind}}\) is

    \({E}_{\text{ind}}=\cfrac{1}{2}{\text{LI}}^{2}\text{.}\)

Glossary

inductance

a property of a device describing how efficient it is at inducing emf in another device

mutual inductance

how effective a pair of devices are at inducing emfs in each other

henry

the unit of inductance; \(1\phantom{\rule{0.25em}{0ex}}\text{H}=1\phantom{\rule{0.25em}{0ex}}\Omega \cdot \text{s}\)

self-inductance

how effective a device is at inducing emf in itself

inductor

a device that exhibits significant self-inductance

energy stored in an inductor

self-explanatory; calculated by \({E}_{\text{ind}}=\cfrac{1}{2}{\text{LI}}^{2}\)

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