Update lecture notes EE1P1

Courses/EE1P1_Electricity_and_Magnetism.typ

@@ -62,3 +62,109 @@
 === 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
+  $
+]