Electric field lines between charged plates and magnetic field patterns

Inductance and inductors

PHYS 301 · Electromagnetic Induction

Inductors resist changes in current by inducing emf. This lesson introduces self-inductance, RL circuits, magnetic energy, and energy density.

Key equations

\mathcal{E}_L=-L\frac{dI}{dt}1\ H=1\ V\,s/AL=\frac{\mu N^2A}{\ell}\mathcal{E}-IR-L\frac{dI}{dt}=0I(t)=\frac{\mathcal{E}}{R}\left(1-e^{-t/\tau}\right)\tau=\frac{L}{R}I(t)=I_0e^{-t/\tau}U_L=\frac{1}{2}LI^2U_C=\frac{1}{2}CV^2u_B=\frac{B^2}{2\mu_0}u_B=\frac{B^2}{2\mu}\mathcal{E}_2=-M\frac{dI_1}{dt}

Learning objectives

Define self-inductance and the henry.
Explain how inductors oppose changes in current.
Analyze current growth and decay in RL circuits.
Calculate energy stored in an inductor.
Describe magnetic energy density and mutual inductance.

Self-induction

A current in a circuit creates a magnetic field. If the current changes, the magnetic flux through the circuit changes, inducing an emf in the same circuit. This is self-induction.

The induced emf is proportional to the rate of change of current:

$mathcal{E}_L=-L rac{dI}{dt}$

The constant $L$ is inductance, measured in henrys:

$1 H=1 V,s/A$

The negative sign reflects Lenz's law: the inductor opposes changes in current.

What an inductor does

An inductor is a circuit element designed to have significant inductance, often a coil of wire. It resists changes in current. If current is increasing, the inductor produces an emf opposing the increase. If current is decreasing, it produces an emf trying to maintain the current.

An inductor does not resist steady current in the same way a resistor does, assuming ideal wire. It resists changes in current.

Inductance of a solenoid

For a long solenoid with $N$ turns, length $ell$ , cross-sectional area $A$ , and core permeability $mu$ , the inductance is approximately

$L= rac{mu N^2A}{ell}$

More turns, larger area, and higher permeability increase inductance. Longer length decreases inductance.

RL circuit growth

Consider a resistor $R$ and inductor $L$ connected in series to a battery of emf $mathcal{E}$ . Kirchhoff's loop rule gives

$mathcal{E}-IR-L rac{dI}{dt}=0$

The current grows as

ight)$$ where the time constant is $$ au= rac{L}{R}$$ The inductor prevents current from jumping instantly to its final value. ## RL circuit decay If the battery is removed and the inductor discharges through the resistor, the current decays as $$I(t)=I_0e^{-t/ au}$$ Again, $$ au= rac{L}{R}$$ The inductor's stored magnetic energy is dissipated as heat in the resistor. ## Energy stored in an inductor Building current in an inductor requires work. The stored magnetic energy is $$U_L= rac{1}{2}LI^2$$ This parallels capacitor energy $$U_C= rac{1}{2}CV^2$$ Capacitors store energy in electric fields; inductors store energy in magnetic fields. ## Magnetic field energy density For a magnetic field in vacuum, the energy density is $$u_B= rac{B^2}{2mu_0}$$ More generally, in a linear material, $$u_B= rac{B^2}{2mu}$$ The total magnetic energy can be found by integrating energy density over space. ## Mutual inductance Changing current in one circuit can induce emf in another nearby circuit. This is mutual induction: $$mathcal{E}_2=-M rac{dI_1}{dt}$$ where $M$ is mutual inductance. Transformers rely on mutual inductance between coils. ## The big idea Inductors oppose changes in current through self-induced emf. Their behavior follows Lenz's law and introduces time dependence into circuits. Inductors store energy in magnetic fields, with energy $ rac{1}{2}LI^2$, and they are essential in filters, power systems, motors, transformers, and oscillating circuits.

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