Energy stored in Magnetic Fields

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Energy stored in Magnetic Fields
In a circuit for flowing current it takes a certain amount of energy to start a current. That energy is equivalent to work must do against the back emf to get the current going. This is a fixed amount, and it is recoverable: you get it back when the current is turned off. It can be regarded as energy stored in the magnetic field.

The work done on a unit charge, against the back emf, in one trip around the circuit is: -
work done is against the emf, not by the emf so negative sign is there. The amount of charge per unit time passing down the wire is I. So the total work done per unit time is

In the starting current : 0
Let final value of current is: I We can get the work done by integrating the last equation over time: Lets generalize for surface and volume currents We know that flux through the loop is: On the other hand, Where P is the perimeter of the loop and S is any surface bounded by V. Thus,

and and therefore or generalize to volume currents: Ampere’s law says: therefore Product rule states that: or because

therefore where S is the surface bounding the volume V But any region larger than this will do just as well, for J is zero out there anyway. For larger the region, we pick the greater is the contribution from the volume integral, and therefore the smaller is that of the surface integral In particular, if we agree to integrate over all space, then the surface integral goes to zero.

therefore magnetic energy formulas are similar to their electrostatic counterparts:

Electrodynamics Before Maxwell
But equation 4 i.e. Ampere’s law is true only for magneto-statics and does not hold for time varying fields. Lets take the divergence on left and right side of eq. 4

The left side must be zero, but the right side, in general, is not.
For steady currents: .J is zero But when we go beyond magnetostatics Ampere's law cannot be right. To generalize the law Maxwell Fixed (modified) Ampere's Law Applying the continuity equation: Use Gauss's law:

This is also called modified Ampere’s law.
He added this term in right side of equation. or This is also called modified Ampere’s law. The first term is due to current flowing in the conductor. Second term due to changing electric field. Maxwell called his extra term the displacement current (Jd):

Displacement current (Jd):
Maxwell postulated that it is not only the current in a conductor that produces a magnetic field, but a changing electric field in vacuum or in a dielectric also produces a magnetic field. i.e. Changing electric field is equivalent to a current which flows as long as the electric field is changing. This equivalent current produces the same magnetic effect as an ordinary current in a conductor. This equivalent current is known as displacement current.

Maxwell's Equations ( in differential form)

Maxwell's Equations ( in integral form)

Physical significance of Maxwell's Equations
First equation represents the Gauss’s law in electrostatics for the static charges, which states that electric flux through any closed hypothetical surface is equal to 1/ times the total charge enclosed by the surface. It states that the net magnetic flux through any closed surface is zero or we can say mono pole can’t exist.

It sates that induced electromotive force around any closed surface is equal to the negative time rate of change of magnetic flux through the path enclosing the surface. Therefore it signifies that an electric field is produced by a changing magnetic flux. This is the modified form of Ampere’s circuital law. It is valid for steady and time varying fields and states that the magnetomotive force around a closed path is equal to the sum of conduction current and displacement current. This signifies that a conduction current as well as a changing electric flux produces a magnetic field.

Maxwell's Equations in Free space In free space:

What is the need to them again for matters?
Maxwell's Equations in Matter Maxwell’s equations in free space are correct. What is the need to them again for matters? The materials which can get electric and magnetic polarization have accumulations of "bound" charge inside the material and current over which is difficult to measure. It would be nice to reformulate Maxwell's equations in such a way as to make equations in terms of only to those sources we control directly: the "free" charges and currents.

We know, from the static case, that an electric polarization P produces a bound volume charge density: In view of all this, the total charge density can be separated into two parts: Gauss's law can now be written as: or Gauss's law in a medium where D (Electric displacement), as in the static case, is given by

Likewise, a magnetic polarization (or "magnetization") M results in a bound volume current
Bound current is associated with magnetization of the material and involves the spin and orbital motion of electrons In the nonstatic case: just one new feature to consider Any change in the electric polarization involves a flow of (bound) charge (call it Jp), which must be included in the total current.

Jp, is the result of the linear motion of charge when the electric polarization changes.
If P points to the right and is increasing, then each plus charge moves a bit to the right and each minus charge to the left; the cumulative effect is the polarization current Jp. Therefore, the current density is: Now the current density is into three parts:

Ampere's law (with Maxwell's term) becomes:
and we know: If Therefore: H is named as auxiliary field. D is electric displacement, therefore is called displacement current

Faraday's law and •B = 0 are not affected by our separation of charge and current into free and bound parts, since they do not involve p or J. In terms of free charges and currents, Maxwell's equations in matters are:

Relation between D and E in matter
In a dielectric medium, polarization (P) occurs due to external field E. We know for linear dielectrics and so where or  and e are permittivity and electric susceptibility of the medium respectively.

Relation between H and B in matter
Similarly for linear media: and so or or where  and m are permeability and magnetic susceptibility of the medium respectively.

Maxwell's Equations in Dielectric medium with

Maxwell's Equations in Conductive medium
In our “simple” conductor, Maxwell’s equations take the form: Where J is the current density. Assuming an ohmic conductor, we can write: with