# The physical meaning of inductance

When we use the term 'inductance', we're usually shortening what should actually be called 'self-inductance'.

When a current flows through a wire, a magnetic field is produced around that wire, with its field strength in proportion to the current strength, changing as the current changes.

We also know that a changing magnetic field will induce a voltage over a wire in proportion to the strength of the field and the rate at which it changes.

So, if we have a wire with a changing current flowing through it (think AC or DC with some ripple), then it will produce a changing magnetic field around itself. This changing magnetic field will then induce a voltage back across that same wire, but this induced voltage will be of the opposite polarity to the voltage of the original applied current.

Since the induced voltage opposes the applied voltage/current, the apparent effect is an increased impedance (complex resistance, or resistance with a phase angle), and since the strength of the induced voltage increases in proportion with the frequency of the applied voltage/current (as frequency is the rate of change), the inductor's impedance increases with frequency.

The inductance (with the unit Henrys) of an inductor is a measure of this self-inductance effect.

(Self)inductance is increased by coiling the wire around so that the coils all contribute to and share the same magnetic field.

Wrapping the coils around a magnetically permeable core also increases (self)inductance by 'concentrating' the magnetic field.

If you expect an answer of the type

"R" of a conductor is how easily the electrons flow through it

(actually it should say how "difficult" instead of "easy")

I'd say

"L" is how difficult it is to change to current through the component.

In addition to the other answers, the symbol for inductance is L, not X. The symbol for inductive reactance is \$X_{L}\$, and for capacitive reactance is \$X_{C}\$.

What is the the physical meaning of inductance?

Hugh Young, in his textbook *University Physics* [1], states the following: for a coil with \$N\$ turns of wire carrying current \$i\$, the current creates a magnetic flux \$\Phi_{B}\$ that passes through each turn. The coil's inductance \$L\$ (a.k.a. *self-inductance*) is given by

$$ L=\frac{N\, \Phi_{B}}{i} $$

Inductors store energy in a field of magnetic flux \$\Phi_{B}\$ whose magnitude is a function of the current flowing through the inductor:

$$ \Phi_{B}=\frac{L\, i}{N} $$

If an electric current flowing through an inductor *changes* with time \$\frac{\mathrm{d} i}{\mathrm{d} t} \neq 0\$, this changing current produces a changing magnetic field \$\frac{\mathrm{d}\Phi_{B}}{\mathrm{d} t} \neq 0\$, that in turn produces a non-zero electromotive force (emf) \$\varepsilon\$ across the inductor, measured in units of *Volts*, whose polarity opposes ("resists") the *change* in current [1]:

$$ \varepsilon = -L\frac{\mathrm{d}i}{\mathrm{d} t} = -N\frac{\mathrm{d}\Phi_{B}}{\mathrm{d} t} $$

This opposition to the *change* in current flow is the *inductive reactance*.

If an electric current flowing through an inductor is constant (does not change with time, a.k.a., *direct current*) \$\frac{\mathrm{d} i}{\mathrm{d} t}=0\$, the field of magnetic flux \$\Phi_{B}\$ is not changing with time (its magnitude is constant) \$\frac{\mathrm{d} \Phi_{B}}{\mathrm{d} t}=0\$, then no emf is produced, \$\varepsilon = 0\$. Therefore, an ideal inductor (zero resistance) does not oppose direct current flowing through it.

When an ideal inductor having inductance \$L\$ is driven with a sinusoidal current having radian frequency \$\omega\$ (or frequency f), the magnitude of the inductor's steady-state inductive reactance \$X_{L}\$ is given by

$$ X_{L}=j\omega L = j2\pi fL $$

where \$j=\sqrt{-1}\$. This equation is derived from a phasor transformation of the first order differential equation

$$ v_{L}(t)=L\, \frac{\mathrm{d}}{\mathrm{d} t}i_{L}(t) $$

where \$v_{L}(t)\$ is the instantaneous voltage across the inductor, and \$i_{L}(t)\$ is the instantaneous sinusoidal current through the inductor at time t:

$$ i_{L}(t) = I_{m} cos(\omega t + \phi) $$

References

[1] H. Young. "Inductance," in *University Physics*, 8th ed. Reading, Massachusetts: Addison-Wesley, 1992, ch. 31, pp. 869-870.