Chapter 5.3, 5.5, and 5.6
Maxwell’s Magnetostatic Equations
Gauss’ Law
$$
\nabla \cdot B = 0, \oint _S B \cdot ds = 0
$$
Ampere’s Law
Electrostatic field is conservative, its line integral along a closed contour always vanishes
$$
\nabla \times H = J, \oint _C H \cdot dl = I \newline \int _s(\nabla \times H) \cdot ds = \int _s J \cdot ds
$$
- The line integral of H around a closed path is equal to the current traversing the surface bounded by that path
- To apply Ampere’s law, the current must flow through a closed path
- Line integral of H is equal to the current I, even though the paths have very different shapes and the magnitude of H is not uniform along the path
Magnetic Field of a Long Wire
Magnetic Field inside a Toroidal Coil
Magnetic Field of an Infinite Current Sheet
Magnetic Properties of Materials
Electron Orbital and Spin Magnetic Moments
$$
m_s = -\frac{eh}{2m_e}
$$
Magnetic Permeability
$$
\mu = \mu _0 (1+ \chi _m) \text{ H/m} \newline \text{or,}
\newline \mu _r = \frac{\mu}{\mu _0} = 1 + \chi _m
$$
Magnetic Boundary Conditions