EM I: Electrostatic Field
Final Review
1. Vector Analysis
- field position vector, source position vector, separation
vector, unit vector, and vector field as a function of
position;
- gradient of a scalar field, divergence and curl of a vector
field (such as a flow field); gradient theorem, Gauss’s
theorem, and Stokes’ theorem;
- curvilinear coordinates: spherical
and cylindrical
r r
coordinates; how to write dl , da, d" in these coordinates;
how to find gradient, divergence and curl, how to compute
path, surface, and volume integrals in these coordinates.
NB: the unit vectors
! in the curvilinear coordinates are NOT
constant vectors.
Ex: calculating the dipole moment of a circular disk
capacitor or spherical shell.
2. Electrostatic field and electric potential
- electric charges: point charge, line charge, surface
charge, and volume charge
- Coulomb’s law: compute electric field and electric
potential by point charges (sum) or continuous charge
distribution (integral).
NB: a. be careful with curly-r in the integral
b. observe symmetry in the distribution
c. apply superposition when appropriate
Ex: calculate electric field or potential by a finite line
charge, surface charge on a circular disk, a volume
charge in a finite length cylinder or a sphere.
- properties of electrostatic field: divergence, curl, and
boundary condition of the field.
3. Methods to find electrostatic field and/or potential
- symmetric charge distribution: use Gauss’s law to find
electric field across the Gaussian surface.
Ex: point charge, sphere or spherical shell, long line,
cylinder or cylindrical shell, large slab or plate, and
combination of above configurations.
- solve Laplace’s equation in space with no volume charge
+ boundary condition to find the electric potential.
2-3D rectangular pipes: sinusoidal & hyperbolic series
2D spherical geometry: power & Legendre polynomials
2D cylindrical geometry: power & sinusoidal functions.
- multipole expansion to find the potential: dipole moment,
dipole field; no curly-r in the solution!
- method of images
NB: superposition is a very useful tool!
4. Electric field in conductors and in dielectric materials
- properties of the electric field, electric potential, and
charge distribution in a conductor; induced charges in an
external field.
- polarization of dielectric materials: bound charges,
displacement field, linear dielectric materials, boundary
condition.
- you use the same methods in 3. to compute electric field
or potential inside/outside materials.
Ex: electric field in a uniformly charged sphere /cylinder
/slab, or a uniformly polarized sphere /cylinder /slab, or a
sphere /cylinder /slab of linear dielectric material in an
otherwise uniform external field.
5. Energy, forces, capacitors
- the energy of an electric field in vacuum and in materials
- the electrostatic force
Ex: energy and force in a capacitor.