## Mathematical technique

### Vector algebra

dot product:

Cross product:

### Differential Calculus

The gradient:

The position vector

Distance between 2 points

The gradient on

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

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

Total moment of dipole under electric field:

### Flux

### Gauss’s law

Using divergence theorem we have

Also we have

### The curl of 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

### 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

Current:

Current density:

Also

Drude model:

where

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