1. Ampere's Circuit Law
- Point or Differential form: The curl of static magnetic field intensity H at any point in the electromagnetic region is equal to volume current density J present at that point.
- Integral Form: The closed line integral of Static magnetic field intensity H, integrated over any closed curve 'C' is always equal to total current enclosed within the closed curve 'C'.
2. Magnetic Field Intensity due to Infinite Line and sheet current
- Line Current: Suppose we have a current of I'' amperes moving through a thin wire. The current flows in the z direction. Recall that we use the right-hand thumb rule which means that we put the thumb of our right hand in the direction of the current and our fingers will go in the direction of the magnetic field.
- Magnetic flux density for infinite line current will be,
- Infinite Current Sheet: Consider a surface current in the x-y plane flowing in the x direction. The magnetic field will be,
- Let's consider doing the integration over a square path of side length b:
- Note that we only integrated the surface current density over the line from 0 to b in y. We did not integrate into z since the surface current density is in units of A/m (i.e. we only need to integrate into one dimension to get the total current enclosed).
3. Magnetic Energy Density
- The magnetic energy density represents magnetic energy stored at a point in the electromagnetic region and gives total magnetic energy per unit volume of the given configuration.
- The magnetic energy density depends upon magnetic field due to given current distribution in the configuration and permeability of the magnetic medium.
- When an inductor is connected to a current source, 'i' increasing/charging from zero I, the total work performed is
4. Magnetic scalar and vector potentials
- The magnetic potential could be scalar V or vector A.
- magnetic scalar potential: the magnetic scalar potential is only defined in a region where J=0.
H = - ∇V
- magnetic vector potential: In constructing an equation for A, we use Gauss's Law of magnetics that ∇.B = 0. We also take advantage of a vector identity that for any vector A,
- Therefore, we will use
5. Magnetic Torque and Dipole Moment
- Torque arises out of a rotational force, i.e., where d is a moment arm to which a force is applied on one end and the other end is connected to a rotating shaft.
- Drilling down a bit, where r = |D| and F = |D|
- There is also a magnetic moment, m that lies along the normal n to the loop,
6. Magnetic Boundary Conditions
- The boundary conditions of the B and H fields at a material boundary μ1 = μ2 parallels the electric field
- Gauss’s law for magnetics says that,
- In words, B is continuous across a boundary. Using Ampère’s law a closed contour at the boundary,
- where n2 is normal at the material interface.
- When finite conductivity is involved, Js = 0 then it is always true that,
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