This is the inductance per length l of a long solenoid having a crosssectional area A. In class, I demonstrated this result by evaluating the flux linkage per unit current of N = nl turns of the solenoid.
ParallelPlate Capacitor  Long Solenoid  
Uniform electric field
Plate area A, 
Uniform magnetic field
Turn density n, 

Electric Energy / Volume  Magnetic Energy / Volume  



Energy/Volume in Capacitor  Energy/Volume in Solenoid  



Capacitance of this Geometry  Inductance of this Geometry  



Electric Energy Density  Magnetic Energy Density  


Although the resulting expressions for the energy density associated with fields has been worked for specific ideal cases involving uniform fields, the formulas for energy density are general results! There is energy associated pointbypoint with both electric and magnetic fields.
When we studied the electric energy density earlier in the course, we used it as an alternative way of evaluating capacitance. Similarly we will be using the magnetic energy density integrated over the volume of an inductor to evaluate its inductance. This is especially useful when the flux linkage is difficult to identify, as in the case of the contribution to inductance arising from the field inside the wires of the inductor...
"http://www.physics.udel.edu/~watson/phys208/clas0422.html"
Last updated April 22, 1998.
Copyright George Watson, Univ. of Delaware, 1997.