Just as the capacitance of a capacitor had both a geometric and an operational definition, so does the inductance of an inductor. Inductance is often referred to as the electrical equivalent of inertia. This property will become more clear when we later investigate LR and LC circuits. An inductor is a device placed in a circuit to oppose a change in current; that is, to maintain, and regulate, a steady current in that section of the circuit. Generally an inductor is thought of as a coil of wire wound around either an air or ferromagnetic core. Shown below is the symbol for an inductor.

 

 

 

Starting with we learned that one farad of capacitance meant that the capacitor had the ability to hold one coulomb of charge for each volt of potential placed across its "plates." The self-inductance (or just inductance) of a coil is presented as where one henry of inductance means that the solenoid produces one weber of magnetic flux for each ampere of current traveling through its loops. Just like one farad is a large amount of capacitance, one henry is a large amount of inductance. Consequently inductors are usually measured in mH, µH, and nH. Continuing our comparison, recall that we used a gaussian surface to derive the capacitance of two parallel plates.

 

 

Now we will use an amperian loop to derive the self-inductance of a solenoid consisting of several turns (or loops) of wire. 

The flux through an area, A, resulting for a perpendicular field, , is expressed as     When current is running through the coils, a uniform magnetic field is produced down the center of the inductor. That is, a given amount of magnetic flux is present. If the current were to change, the amount of flux would change. This changing flux induces an opposing emf in the coil. This self-induced emf is sometimes called a back emf.

 

Notice that L, the inductance, is proportional to the square of the number of loops, directly proportional to the solenoid's cross-sectional area, and inversely proportional to the length of the solenoid. Recall that µ_o is the permeability of free space. If a ferromagnetic cylinder is inserted into the core of the solenoid, the µ = κu_o. Some typical values for relative permeabilities are:

• air 1.00000037
• nickel 100
• magnetic iron 200

 

One last note. Ideal solenoids have a perfect flux-linkage - that is, the flux through any coil will encircle ALL of the coils. In a real solenoid this only occurs when "the cross-sectional dimension of the windings are small compared to the coil's diameter or if a high permeability core is inserted to guide the flux" completely through all of the windings. In a REAL air-core solenoid, the flux density tends to decrease towards the ends of the coil since some of the flux lines "cut through" instead of encircle the final coils. Hence why we experimentally are always directed to measure the B-field of a solenoid in the center of its middle coils.

 

Quote and image used with the permission of Richard A. Clark
from his webpage http://info.ee.surrey.ac.uk/Workshop/advice/coils/terms.html#link

 

 

Faraday's Law states the emf induced is a coil is proportional to the rate of change of flux. Later, Lenz added that the induced emf will be established in such a way as to oppose these changes and return the coil to its original condition. (This is actually a statement of conservation of energy. If the flux generated by the coil did NOT oppose the change in external flux then the induced emf would continue without limit.)

 

 

Substituting in our previous results we can develop an expression for the emf induced in an inductor.

 

 

Notice that the faster the current tries to change, the greater the inductor's "back emf." When the current is steady, , there will be NO back emf in the solenoid, or inductor.

 

 

 

We will now use a graph of Flux vs Current to further examine the properties of an inductor.

 

The magnetic energy density is the energy per unit volume stored in a magnetic field.