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\begin{document}
\title{Electricity}
\author[1]{Ricva E Hildebrandt}%
\affil[1]{The Hebrew University of Jerusalem}%
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\date{\today}
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\section*{Formula sheet}
\section*{Electrostatics}
\subsection*{Coulomb's constant}
It was introduced by Coulomb for the Coulomb's law. It can be defined as:\\
\(k\ =\ \frac{1}{4\pi\epsilon_0}\)\\
\(k\ =\ 9\cdot10^9\frac{Nm^2}{C^2}\)\\
where:\\
\(epsilon_0\) is the vacuum permitivity\\
\subsubsection*{Vacuum permitivity}
It is also defined as the distributed capacitance of the vacuum. In general, permitivity is the capacitance encountered when a electric field is formed in a certain medium. The vacuum permitivity is:\\
\(\epsilon_0\ =\ \frac{1}{4\pi K}\ =\ 8.84\cdot20^{-12}\)\\
\subsection*{Coulomb's force}
It quantifies the amount of force with which stationary charges repel or attract each other.\\
\(F_{12}\ =\ k\frac{Q_1Q_2}{r_{12}^2}\hat{r}_{12}\)\\
In other words, the force applied by one stationary point charge on another is equals to the multiplication of their charges, divided by the distance between them and multiplied by the Coulomb's constant \(k\). This force is radial, i.e. it acts on the same direction as the vector distance from one particle to the other.
\subsection*{Electric field}
It is defined by a vector field surrounding a charge that exerts electric force on other charges. Mathematically, it is defined by:\\
\(\vec{E}\ =\ \frac{\vec{F}}{q}\)\\
In other words, the electric field is the electric force divided by a unit charge, or how much force is applied to a single charge and in which direction.\\
\(\vec{E}\ =\ k\int\frac{\rho\left(r'\right)\left(r-r'\right)}{\left|r-r'\right|^3}dV'\)\\
where:\\
\(\rho\) is the volumetric charge density\\
The electric field is always in the radial direction.
\subsection*{Gauss law}
It has a differential and an integral form. In its differential form, it connects the divergence of the electric field to the volumetric charge density. It means that the amount of compression or decompression of the electric field is equal to the volumetric charge density divided by the vacuum permitivity.\\
\(\bigtriangledown\cdot\vec{E}\ =\ \frac{\rho}{\epsilon_0}\)\\
In its integral form, it relates the electric flux to the ratio of the inside charge to the vacuum permitivity.The electric flux measures the flow of the electric field through a certain area.\\
\(\Phi_{\vec{E}}=\ \oint\vec{E}\cdot d\vec{S}\ =\ \frac{1}{\epsilon_0}\int\int\int\rho\left(r'\right)dV'\ =\ \frac{Q_{in}}{\epsilon_0}\)\\
The first surface integral means the flux is equal to the how much electric field there is in a small element of the area. In the second part of the identity, the triple integral represents the amount of charge in the volume and it comes from:\\
\(\rho\ =\ \frac{Q_{in}}{V}\)\\
\(Q_{in\ }=\ \rho V\)\\
\(dQ_{in}\ =\ \rho dV\)\\
\(\int_0^{Q_{in}}dQ'_{in}\ =\ \int\rho dV\)\\
It's important to note that:\\
\(\bigtriangledown\times\vec{E}\ =\ 0\)\\
Namely, the compression or decompression of a electrical field is adirectional and a electric field is always conservative.
\subsection*{Electric potential}
Electric potential is a scalar field. It represents the amount of work needed in order to move a point charge from a reference point to a specific point inside the electric field.\\
\(\vec{E}\ =\ -\bigtriangledown\phi\)\\
\(\phi\ =\ -\int_{ }^{ }\vec{E}\cdot d\vec{l}\)\\
In plain English, a electric field can be defined as the gradient of the electric potential. And the potential is the integral of the electric field. Another way to define electric potential is:\\
\(\phi\left(\vec{r}\right)\ =\ k\int\int\int\frac{\rho\left(r'\right)}{\left|r-r'\right|}dV'\)\\
\subsection*{Potential energy contained in a electric field}
The reasoning here goes in a similar way as the mechanical potential energy, being the electric potential energy the amount of work needed to bring a point charge from a reference place to a point in the electric field.
\(U\ =\ k\frac{q_1q_2}{r}\)\\
\(U\ =\ -W_{r_{ref}\rightarrow r}\ =\ q\Phi\)\\
\subsection*{Capacitance}
It's how much electric charge the body can hold on a certain voltage. In other words, capacitance is equals to electric charge divided by the difference in potential. And that's what can be seen in the formula: \(C\ =\ \frac{Q}{V}\)
\subsubsection*{Plate capacitor}
Two plates arranged in parallel and connected to a voltage source will form a capacitor. The way to calculate its charge storage capacity, i.e. its capacitance is: \(C\ =\ \epsilon\frac{S}{d}\). It takes into consideration the permitivity (and this is \(\epsilon\))of the material between the plates, because it affects the transfer of electric charges between them. The area of the plate, \(S\), and the distance between the plates is also considered. In other words, the capacitance of a plate capacitor is the relation between the are of the plate and the permitivity of the material in an inverse relation to the distance between the plates.
\subsubsection*{Energy stored in a capacitor}
The energy stored on a capacitor is the integral of the voltage over the charge. The amount of energy contained in the capacitor is the quantity of free charge in relation to the charge stored by the capacitor. The formula is: \(\int_0^QVdq\ =\ \int_0^Q\frac{q}{C}dq\ =\ \frac{1}{2}\frac{Q^2}{C}\), from the capacitance formula: \(C\ =\ \frac{Q}{V}\ \Rightarrow V\ =\ \frac{Q}{C}\). Then, the energy stored in a capacitor is: \(U\ =\ \frac{1}{2}\frac{Q^2}{C}\)
\subsection*{Dipole}
The electric dipole formed by point charges is a vector given by a scalar multiplication between the amount of charge and the distance between them. Its formula is: \(\vec{p}\ =\ q\vec{d}\) The dipole formed by continuous charge is given by the integral of the charge density over the volume. The formula is: \(\vec{p}\ =\int_{ }^{ }\vec{r}\ \rho\left(r\right)dV\).
The difference in potential between the edges of the dipole is given by: \(\varphi\left(r\right)\ =\ K\frac{\vec{r}\cdot\vec{p}}{r^3}\). And the electric field is: \(\vec{E}\ =\ K\frac{3\left(\vec{p}\cdot\ r\right)r\ -\ \vec{p}}{r^3}\)
\subsection*{Polarization}
\subsubsection*{Bound charge}
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