1. 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:

2. 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:

3. 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:

4. 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:

5. 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.

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