Energy in magnetic fields When a electric current is flowing in an inductor, there is energy stored in the magnetic field. Considering a pure inductor L, the instantaneous power which must be supplied to initiate the current in the inductor is. so the energy input to build to a final current I is given by the integral Let us write the energy in a more general way. arbitrary loop:

energy density of the magnetic field: Remember that Hence, we can generalize the above expression to an arbitrary distribution of volume currents Vanishes at large distance from the sources energy density of the magnetic field: reminder energy density of the electric field:

very thin metallic shield of radius b centre core of radius a Example: Find magnetic energy per unit length stored in a coaxial cable carrying current I. I I very thin metallic shield of radius b centre core of radius a According to the Ampere’s law, the field is nonzero only in the area between the conductors (a<r<b). This immediately yields:

Maxwell’s Equations Gauss’s law no magnetic monopoles! Faraday’s law So far, we have encountered the following laws: Gauss’s law no magnetic monopoles! Faraday’s law Ampere’s law Problem! only for steady currents. The problem was solved by Maxwell, who used the continuity equation:

Ampere’s law is cured if we replace displacement current A changing electric field induces a magnetic field Integral version of generalized Ampere’s law: An electrically charging capacitor with a Gaussian cylindrical surface surrounding the left-hand plate. Right-hand surface R lies in the space between the plates and left-hand surface L lies to the left of the left plate. No conduction current enters cylinder surface R, while current I leaves through surface L. Consistency of Ampère's law requires a displacement current Id = I to flow across surface R.

Maxwell’s Equations Gauss’s law no magnetic monopoles! Faraday’s law Ampere’s law with Maxwell’s correction