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Ampère's Law

You are probably also adept at using the trick developed by Henri Ampère for calculating the magnetic field ( $\Vec{H} \equiv \Vec{B}/\mu$) due to various symmetrical arrangements of electric current (I). In its integral form and SI units, AMPÈRE'S LAW reads

$$\oint_{\cal C} \, \Vec{H} \cdot d\Vec{\ell} \; = \; I + {\p . . . . . . over \partial t} \SurfInt_{\cal S} \, \Vec{D} \cdot d\Vec{S}$$ (22.7)

where Maxwell's " DISPLACEMENT CURRENT" associated with a time-varying electric displacement   $\Vec{D} \equiv \epsilon \Vec{E}$  has been included. This equation says (sort of), "The circulation of the magnetic field around a closed loop is equal to a constant times the total electric current linking that loop, except when there is a changing electric field in the same region."

As you know, this "Law" is used with various symmetry arguments to "finesse" the evaluation of magnetic fields due to arrangements of electric currents, much as GAUSS' LAW was used to calculate electric fields due to different arrangements of electric charges. Skipping over the details, let me draw your attention to the formal similarity to FARADAY'S LAW and state (this time without showing the derivation) that there is an analogous differential form of AMPÈRE'S LAW describing the behaviour of the fields at any point in space:

$$\hbox{\fbox{ ${\displaystyle \Curl{H} \; = \; \Vec{J} \; + \; {\partial \Vec{D} \over \partial t} }$\space } }$$ (22.8)

If we ignore the current density  $\Vec{J}$  then this equation says (sort of), "A changing electric field generates a magnetic field at right angles to it," which is rather reminiscent of what FARADAY'S LAW said.

Now we're getting somewhere.