Mathematical technique

Vector algebra

dot product:

Cross product:

Differential Calculus

The gradient:

The position vector $\va r$:

Distance between 2 points $\va r$ and $\va r'$:

The gradient on $|\va r - \va r'|$:

The divergence:

The curl:

Product rules:

2 for gradients/curls/divergences:

2nd divergences (def of vector Laplacian):

Suffix notation

Tensor (notations in this document):

In the operation of tensors, it is often to see the summation of the dummy subscript twice in any term of an expression with a $\sum$.

dot product (is a scalar):

Tensor operation (is a vector):

The Kronecker delta:

Permutation symbol:

Cross product:

grad, div, curl:

Some useful identities:

Representations of lines, planes, and surfaces using vectors

line:

plane:

Line element:

Surface element:

Integral Calculus

Line integral:

Surface integral:

The fundamental theorem for gradients:

The divergence theorem:

Stokes theorem:

3D Dirac delta function:

Electrostatics

Coulomb’s law

The electric field of point charge:

The electric field of charge distributions:

In 3D:

In 2D:

In 1D:

Dipole

Electric dipole moment from $-q$ to $q$:

Total moment of dipole under electric field:

Flux

Gauss’s law

Using divergence theorem we have

Also we have

The curl of $\va E$ in electrostatics (conservative)

Electric potential:

For a point charge:

Electrostatic potential energy:

Poisson equation

Work and conductors

Capacitors

Parallel plate capacitor:

Spherical metal shells between radii of $a$ and $b$:

Work done to remove a point charge

Therefore,

Electrostatic energy

The energy of a point charge distribution:

Case of 1 charge:

Case of 2 charges:

Case of 3 charges:

Case for many point charges:

The energy of a continuous charge distribution:

Conductor

The method of images…

Induces surface charges:

Magnetostatics

Current and current density

Drift velocity:

where $1/\tau$ is the scatting rate.

Current:

Current density:

Also

Drude model:

where $\sigma$ is the conductivity.

Ohm’s law:

Resistance:

Resistivity:

Magnetic field

Lentz force law:

Cyclotron formula:

Magnetic force on current:

Magnetic force on charge / current density:

Conservation of charges (using the divergence theorem):

The Bict-Savart’s law:

The divergence and curl of the B-field

Ampere’s law:

By using Stokes’s theorem,

Gauss’s law for magnetism:

Some applications of Ampere’s law

For a coil:

For a wire:

EM induction

Electromotive force (e.m.f.):

where

Magnetic flux

Therefore,

Faraday’s law of induction

Faraday’s law (by taking stokes theorem):

 

 

Last Updated on 2 years by Yichen Liu