For vacuum and material with constant susceptibility M 21 is a constant and given by Inductance We know already: changing magnetic flux creates an emf changing current in a coil will induce a current in an adjacent coil Current i 1 through coil 1 creates B-field and thus flux through coil 2. If i 1 =i 1 (t) then dB/dt  0 and thus d  /dt  0 inducing an emf in coil2. Coupling between coils is described by mutual inductance mutual inductance From Using

with For the vacuum case ( but also in general ) mutual inductance depends only on coil geometry with Mutual inductance is measured in Henry where 1H=1Wb/A=1Vs/A=1Ωs=1J/A 2 - Mutual inductance can give rise to cross-talk in electronic circuits  - Mutual inductance has important applications, e.g., in transformers

Many applications happen in AC circuits (see textbook). Here a collection: Transformers Metal detectors Here we have a brief look at the Tesla coil Current i 1 through coil 1 creates B-field and flux solenoid 1 is long compared to solenoid 2 M depends only on geometry and in particular on product N 1 N 2.

To see how a Tesla coil can create a vary large emf let’s have a look to an example Drive a current i 2 (t)through solenoid 2 (blue) Flux through solenoid 1 is given by emf induced in solenoid 1 reads: Hallmark of Tesla coil is the loose coupling (large air gap) between the solenoids 1 and 2 to prevent damage (insulation between tightly coupled solenoids would experience dielectric breakdown). From Note, I did not check the scientific validity of the information provided on this web-site

Self-Inductance and Inductors The concept that a changing flux induces an emf can also be applied in the case of a single solenoid Self-inductance With A device designed to have a particular inductance L is called an inductor Examples from Circuit symbol

Note: Sign opposite to emf The effect of an inductor in a circuit Let’s compare resistor and inductor Emf Current flowing through resistor R gives rise to potential drop opposes current change Potential drop Example: Inductance of an air core toroidal solenoid 1) Determine B from Ampere’s law r 2) Determine flux through one loop 3) Determine emf of solenoid and compare with

Magnetic field Energy We will see that similar to the electric field there is energy stored in a magnetic field Let’s calculate the energy input U needed to establish a current I in an ideal (zero resistance) inductor with inductance L Withwe obtain for the power, P, delivered to the inductor Energy stored in an Inductor For the energy delivered after time, t, we obtain Changing integration variable from t to i we obtain Energy stored in an inductor when permanent current I is flowing

We now want to use to see that the energy is stored in the field ( very much in analogy to the transition from energy in a capacitor to energy stored in the electric field ) Let’s recall the inductance L of a toroidal inductor Volume, V, which is filled with a magnetic field of magnitude where A is the area of the cross-section The energy can be expressed as for i=I which yields the energy density u=U/V for vacuum orin a magnetic material

R-L circuits Kirchhoff’s loop rule t=0 is time when switch is closed all the voltage drops across L and thus i(t=0)=0 Solving the differential equation:

i t What happens if we release energy stored in the solenoid Kirchhoff’s loop rule: i t I 0 /e

L-C circuits L-C circuit shows qualitative new behavior Because there is no power dissipation, energy once stored in C or L will periodically redistribute between energy in E-field and B-field From Kirchhoff’s loop rule Compare with harmonic oscillator