171 lines
3.8 KiB
Plaintext
171 lines
3.8 KiB
Plaintext
#import "@preview/cetz:0.4.2": canvas, draw
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#import "@preview/cetz-plot:0.1.3": plot
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#import "../template/lib.typ": *
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#set page(paper: "a4")
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#show: notes.with(
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title: [EE1P1],
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subtitle: [Electricity and Magnetism],
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author: "Folkert Kevelam"
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)
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= Lecture 1 - Mathematical Instruments
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== Vectors
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#definition[
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definition for a vector with a given source point by $arrow(r)_1$
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$
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arrow(r)_(1,2) &= (x_2-x_1)hat(x) + (y_2-y_1)hat(y) + (z_2-z_1)hat(z) \
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|arrow(r)_(1,2)| &= sqrt((x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2) \
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hat(r)_(1,2) &= arrow(r)_(1,2) / (|arrow(r)_(1,2)|)
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$
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Most of the time, vectors are defined from a source point to a observation
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point.
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$
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arrow(R) = arrow(r) - attach(limits(r), t: arrow prime)
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$
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]
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#definition[
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Scalar product: $RR^3 dot RR^3 arrow RR$, $arrow(a)dot arrow(b) = arrow(b) dot arrow(a)$
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Cross product: $RR^3 times RR^3 arrow RR^3$, $arrow(a) times arrow(b) = -arrow(b) times arrow(a)$
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$arrow(a) times arrow(b) = ||$
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$|arrow(a) times arrow(b)| = |arrow(a)||arrow(b)| sin(alpha)$
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$arrow(a) dot (arrow(b) times arrow(c)) = arrow(c) dot (arrow(a) times arrow(b))$
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$arrow(a) times (arrow(b) times arrow(c))$
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]
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== Integrals
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=== Integrals over $cal(D) subset RR^3$
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+ Compact form: $integral_cal(D) f(x,y,z) d V$
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+ Extended form: $integral.triple_cal(D) f(x,y,z) d V$
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=== Polar coordinates
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+ EE1M1: $r, theta$, EE1P1: $rho, phi$
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=== Cylindrical coordinates
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+ EE1M1: $r, theta, z$, EE1P1: $rho, phi, z$
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=== Spherical coordinates
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+ EE1M1: $r, theta, phi$, EE1P1: $rho, phi, theta$
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= Lecture 2
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= Seminar 1 - Coulomb force
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== Coulomb force
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Force between two charges:
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$
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&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) \
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&arrow(r)_(S O) = arrow(r)_O - arrow(r)_S \
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&r_(S O) = |arrow(r)_(S O)| = |arrow(r)_O - arrow(r)_S| \
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&hat(r)_(S O) = arrow(r)_(S O) / r_(S O) \
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&k = 10^(-7) c_o^2 approx 8.99 dot 10^9 "N sym.dot m"
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$
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= Lecture 3
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*polarisation* is when an electrical field is applied to a dielectric, which
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causes the dipoles within the dielectric to orient to the same point, along
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the external electric field.
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= Lecture 4 - Gauss's law
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== Gauss's law in integral form
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Based on the idea of *flux*:
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- *flux* is the idea of how much of "something" crosses a given surface
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- The *electrostatic field* does not cross anything (it was originally believed to)
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The electric flux can be calculated using the surface integral of the electric field.
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#definition[
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electric flux:
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$
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Phi = integral_cal(S) arrow(E) dot d arrow(A) = integral_S E cos(theta) d A
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$
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for a uniform field and a flat surface:
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$
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Phi = integral_cal(S) arrow(E) dot d arrow(A) = E A cos(theta)
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$
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]
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#definition[
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Gauss's law:
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$
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integral.cont_cal(C) arrow(E) dot d arrow(A) = q_("enclosed") / epsilon_0
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$
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Gauss's law in a medium:
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$
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integral.cont_cal(C) dot d arrow(A) = q_("enclosed") / (epsilon_0 epsilon_r)
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$
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]
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assuming that homogenisation is applicable. \
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Gauss's law is useful for calculating the electric field in symmetric configurations:
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- spherical symmetry
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- cylindrical symmetry
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- symmetry with respect to a plane
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$
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arrow(E)(arrow(r)) = arrow(E)(r, theta, phi) \
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arrow(E)(arrow(r)) = E_r hat(r) \
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$
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*Two distinct cases*:
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+ Observation point is inside the surface.
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with a sphere, with a uniform charge density the total charge is:
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$
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Q_("tot") = rho (4/3) pi R^3
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$
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+ Observation point is outside the surface.
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=== Examples
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==== Calculating the electric field in a sphere with a single charge
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==== uniformly charged rod
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== Conditions at the surface of perfect conductors
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== Local form of Gauss's law
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- The function describing $arrow(E)$ is continuously differentiable.
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- the medium can be taken as "continuous" (homogenisation).
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#definition[
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Gauss's law in local-form:
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$
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arrow(nabla) dot arrow(E)(arrow(r)) = rho(arrow(r)) / epsilon_0
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$
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]
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