Electricity and magnetism

COULOMB's Law

\[ F_{\mathrm{el}} = k \frac{q_{1}q_{2}}{r^{2}}; \; k = \frac{1}{4 \pi \varepsilon_{0}} \]

\[ F_{\mathrm{el}} = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q_{1}q_{2}}{r^{2}} \]

\[ q \Rightarrow Coulomb \, [C] \]

\[ e = 1.6 \cdot 10^{-19} (C) \]

electric field

\[ \vec{F}_{\mathrm{el}} = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q_{0}q}{r^{2}} \vec{u}_{r} \]

\[ \vec{E} = \frac{\vec{F}_{\mathrm{el}}}{q_{0}} = \frac{1}{4 \pi \varepsilon_{0}} \cdot \frac{q}{r^{2}} \vec{u}_{r} \]

electric field lines: 1

electric dipole

electric quadrupole

electric octupole

ELECTRIC FIELD LINES: 2

symmetric dipole

asymmetric dipole

ELECTRIC FIELD LINES: 3

CONTINUOUS distributions of charges

chared plates

\[ | \vec{E} | \approx constants \]

\[ \vec{a} = \frac{\vec{F}}{m} = \frac{q \vec{E}}{m} \]

facing charged plates

moto in un campo elettrico tra due armature cariche

flux of a vectorial field

\[ \Phi = \vec{E} \cdot \vec{A} = E \, A \, cos(\theta) \]

\[ \Phi = \sum_{i} \vec{E}_{i} \cdot \Delta \vec{A}_{i} \Rightarrow \Phi = \int_{A} \vec{E} \cdot d\vec{A} \]

gauss's LAW

"The net electric flux through a closed surface is proportional net electric charge enclosed within that closed surface"

\[ \Phi_{E} = \int_{A} \vec{E} \cdot d\vec{A} = \frac{q_{tot}}{\varepsilon_{0}} \]

gauss'S law examples

electric potential

"The electrostatic force is a conservative force, therefore its work can be expressed through a variation of potential energy"

\[ L^{el}_{A \to B} = \int_{A}^{B} \vec{F}_{el} \cdot d \vec{s} = -(U_{e}(B) - U_{e}(A)) = - \Delta U_{e} \]

\[ \frac{L^{el}_{A \to B}}{q} = \int_{A}^{B}\vec{E}_{el} \cdot d \vec{s} = -(V(B) - V(A)) = - \Delta V \]

\[ \Delta U_{e}= q \Delta V \]

\[ U_{e} \to [J] \; \Rightarrow \; V \to \left[ \frac{J}{C} \right] ;\; [Volt] \]

potential of a positive charge

potential of a negative charge

potential of a charge distribution

electrical current

\[ I = \lim_{\Delta t \to 0} \frac{\Delta q}{\Delta t} = \frac{dq}{dt} \]

\[ Ampere \, [A] \Rightarrow \left[ \frac{C}{s} \right] \]

electrical current: microscopic interpretation

\[ I = q \,n \, A \, v_{d} \]

\( q = carrier \, charge \, [C] \)
\( n = free \, carriers \, for \, unit \, volume\, [m^{-3}] \)
\( A = conductor \, cross \, section \, [m^{2}] \)
\( v_{d} = drift \, velocity \, [m/s]; \, v_{d} \propto \Delta V \)

\[ J = \frac{I}{A} = q \, n \, v_{d} = current \, density \]

ohm's law:

RESISTANCE, RESISTIVITY, CONDUCTIVITY AND MOBILITY

\begin{cases} R = \rho \frac{L}{A}, \\ \sigma=\frac{1}{\rho} \\ \sigma=q n \mu \\ \end{cases}

\begin{cases} V = IR \\ \vec{E}=\rho \vec{J} \\ \vec{J}=\sigma \vec{E} \\ \vec{v_d} = \mu \vec{E} \end{cases}

\begin{cases} R = resistance \Rightarrow Ohm \, [\Omega] = \left[ \frac{V}{A} \right] \\ \rho = resistivity \Rightarrow \left[ \frac{\Omega}{m} \right] \\ \sigma = conductivity \Rightarrow \left[ \frac{m}{\Omega} \right] \\ \mu = mobility \Rightarrow \left[ \frac{m^2 s^{-2}}{V} \right] \\ \end{cases}

magnetism

comparing electric and magnetic field

\[\vec{F}_{el} = q \vec{E} \]

\[ \vec{F}_{mag} = q \, (\vec{v} \times \vec{B}) \]

biot-savart's law

\[ d \vec{B} = \frac{\mu_{0}}{4 \pi} \frac{i \, d \vec{s} \times \vec{u}_{r} }{r^{2}} \]

ampere's law

\[ \oint_S \vec{B} \cdot d \vec{s} = \mu_{0} i \]