#import "@preview/cetz:0.4.2": canvas, draw
#import "@preview/cetz-plot:0.1.3": plot

#import "../template/lib.typ": *
#set page(paper: "a4")
#show: notes.with(
  title: [EE1P1],
  subtitle: [Electricity and Magnetism],
  author: "Folkert Kevelam"
)

= Lecture 1 - Mathematical Instruments

== Vectors

#definition[
  definition for a vector with a given source point by $arrow(r)_1$
  $
    arrow(r)_(1,2) &= (x_2-x_1)hat(x) + (y_2-y_1)hat(y) + (z_2-z_1)hat(z) \
    |arrow(r)_(1,2)| &= sqrt((x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2) \
    hat(r)_(1,2) &= arrow(r)_(1,2) / (|arrow(r)_(1,2)|)
  $

  Most of the time, vectors are defined from a source point to a observation
  point.

  $
    arrow(R) = arrow(r) - attach(limits(r), t: arrow prime)
  $
]

#definition[

  Scalar product: $RR^3 dot RR^3 arrow RR$, $arrow(a)dot arrow(b) = arrow(b) dot arrow(a)$

  Cross product: $RR^3 times RR^3 arrow RR^3$, $arrow(a) times arrow(b) = -arrow(b) times arrow(a)$

  $arrow(a) times arrow(b) = ||$

  $|arrow(a) times arrow(b)| = |arrow(a)||arrow(b)| sin(alpha)$

  $arrow(a) dot (arrow(b) times arrow(c)) = arrow(c) dot (arrow(a) times arrow(b))$

  $arrow(a) times (arrow(b) times arrow(c))$
]

== Integrals

=== Integrals over $cal(D) subset RR^3$

+ Compact form: $integral_cal(D) f(x,y,z) d V$
+ Extended form: $integral.triple_cal(D) f(x,y,z) d V$

=== Polar coordinates

+ EE1M1: $r, theta$, EE1P1: $rho, phi$

=== Cylindrical coordinates

+ EE1M1: $r, theta, z$, EE1P1: $rho, phi, z$

=== Spherical coordinates

+ EE1M1: $r, theta, phi$, EE1P1: $rho, phi, theta$

= Lecture 2

= Seminar 1 - Coulomb force

== Coulomb force

Force between two charges:

$
  &arrow(F)_(S O)(arrow(r)_(S O), Q_S, Q_O) = k (Q_S Q_O) / r_(S O)^2 hat(r)_(S O) \
  &arrow(r)_(S O) = arrow(r)_O - arrow(r)_S \
  &r_(S O) = |arrow(r)_(S O)| = |arrow(r)_O - arrow(r)_S| \
  &hat(r)_(S O) = arrow(r)_(S O) / r_(S O) \
  &k = 10^(-7) c_o^2 approx 8.99 dot 10^9 "N sym.dot m"
$

= Lecture 3

*polarisation* is when an electrical field is applied to a dielectric, which
causes the dipoles within the dielectric to orient to the same point, along
the external electric field.

= Lecture 4 - Gauss's law

== Gauss's law in integral form

Based on the idea of *flux*:

- *flux* is the idea of how much of "something" crosses a given surface
- The *electrostatic field* does not cross anything (it was originally believed to)

The electric flux can be calculated using the surface integral of the electric field.

#definition[
  electric flux:

  $
    Phi = integral_cal(S) arrow(E) dot d arrow(A) = integral_S E cos(theta) d A
  $

  for a uniform field and a flat surface:

  $
    Phi = integral_cal(S) arrow(E) dot d arrow(A) = E A cos(theta)
  $
]

#definition[
  Gauss's law:

  $
    integral.cont_cal(C) arrow(E) dot d arrow(A) = q_("enclosed") / epsilon_0
  $

  Gauss's law in a medium:

  $
    integral.cont_cal(C) dot d arrow(A) = q_("enclosed") / (epsilon_0 epsilon_r)
  $
]

assuming that homogenisation is applicable. \
Gauss's law is useful for calculating the electric field in symmetric configurations:

- spherical symmetry
- cylindrical symmetry
- symmetry with respect to a plane

$
  arrow(E)(arrow(r)) = arrow(E)(r, theta, phi) \
  arrow(E)(arrow(r)) = E_r hat(r) \
$

*Two distinct cases*:

+ Observation point is inside the surface.
  with a sphere, with a uniform charge density the total charge is:
  $
    Q_("tot") = rho (4/3) pi R^3
  $
+ Observation point is outside the surface.
  

=== Examples

==== Calculating the electric field in a sphere with a single charge

==== uniformly charged rod



== Conditions at the surface of perfect conductors

== Local form of Gauss's law

- The function describing $arrow(E)$ is continuously differentiable.
- the medium can be taken as "continuous" (homogenisation).

#definition[
  Gauss's law in local-form:

  $
    arrow(nabla) dot arrow(E)(arrow(r)) = rho(arrow(r)) / epsilon_0
  $
]