# 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 $${E}_{\text{ind}}=\cfrac{1}{2}{\text{LI}}^{2}$$