# Inductors

##### Warning: This educational article is under development.

Oersted's law states that when a constant current flows through a conductor, an electromagnetic field is induced around it [1]. When the current is varied with time, the change in the electromagnetic field is opposed by an electromagnetic force (EMF) which resists the change. The EMF is generated in both the conductor itself, and in any other conductors within the magnetic field. This resistance to current change effect is called inductance[2] and is measured in Henries. A conductor (inductor) would have a capacitance of one henry if a change in one ampere per second in the current caused an EMF resistance of 1 volt.

The EMF, a voltage $V$ across the conductor caused by the electromagnetic field, is described as:

$$V = L\frac{dI}{dt}$$

where L is the inductance, measured in Henries, $\frac{dI}{dt}$ is the time rate of change of the current $I$.

### Describing the magnetic field around a conductor (Biot-Savart law)

The magnetic field, $\mathbf{B}(\mathbf{r})$, at a point $\mathbf{r}$ in space around a flowing current is described by the Biot-Savart law [3] as:

$$\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int_C \frac{I d\mathbf{l} \times \mathbf{r}^\prime}{|\mathbf{r}^\prime|}$$,

where $\mu_0$ is the magnetic constant ($4\pi \times 10^{-7}$ N/A$^2$), $I$ is the current in Amperes, $d\mathbf{l}$ is the differential length of the conductor, $\mathbf{r}^\prime = \mathbf{r} - \mathbf{l}$ is the displacement vector between the point $\mathbf{l}$ on the conductor's path and the point of interest $r$, and $C$ is the conductor continuum being integrated.

*Figure 1: An illustration of the magnetic field around a conductor.*

### Coil

If a conductor is coiled such that it runs parallel to itself, the EMF caused by magnetic flux is multiplied by the number of turns of the coil since the EMF created in each turn is added in series. Therefore, the number of loops in a coil has a practically linear effect on the inductance of the coil.

[1] Wikipedia - Oersted's law

[2] Wikipedia - Inductance

[3] Wikipedia - Biot-Savart Law

[4] Wikipedia - Electromagnetic Coil