Electric charge

• Electric charge is quantized, that is, exists in discrete quantities which are integer multiples of the elementary charge $e=1.6022\times 10^{-19}\mathsf{C}\approx 1.6\times 10^{-19}\mathsf{C}$
  • The charge of an electron is $-e$ and the charge of a proton is $+e$
  • The SI unit of charge is the coulomb (C)
• Conservation of charge: the total charge in an isolated system remains constant
• An object can become charged by:
  • rubbing (friction)
  • conduction (transfer of charge from one charged object to another by touching)
  • induction
• $m_{\mathrm{e}}=9.11\times 10^{-31}\mathsf{kg}$ is the mass of an electron
• $n=\frac{Q}{e}$ is the number of electrons transferred

Coulomb’s law

• $F=k\frac{q_1{q}_2}{r^2}$ is the electrostatic force (or Coulomb force) between two charges (in $\mathrm{N}$)

  • $q_1$ and $q_2$ are the magnitudes of the charges (in $\mathrm{C}$)
  • $r$ is the distance between the charges (in $\mathrm{m}$)
  • $k=\frac{1}{4\pi {ϵ}_0}=8.99\times 10^9\mathsf{N}\cdot {\mathsf{m}}^2/{\mathsf{C}}^2$ is Coulomb’s constant

• limitations and assumptions of Coulomb’s Lawtodo

  • point charges
  • objects are at rest (electrostatics force)
  • electric force

• (Superposition principle) The total force on a charge $Q$ is the sum of the forces exerted by the other charges $q_1,q_2,q_3,...$ on $Q$

  • $F=F_1+F_2+F_3+...$

• $|{\vec{F}}_E|=\frac{1}{4\pi {ϵ}_0}\frac{|q_1{q}_2|}{r^2}=k\frac{|q_1{q}_2|}{r^2}$todo

Charge density

• $\lambda =\frac{Q}{L}$ is the linear charge density (in $C/m$)

• $\sigma =\frac{Q}{A}$ is the surface charge density (in $C/{\mathrm{m}}^2$)

• $\rho =\frac{Q}{V}$ is the volume charge density (in $C/{\mathrm{m}}^3$)

  • total charge density: $\rho ={\rho }_f+{\rho }_b={ϵ}_0(\nabla \cdot \vec{\mathbf{E}})$
  • free charge density ${\rho }_f=\nabla \cdot \vec{\mathbf{D}}$
  • bound charge density ${\rho }_b=-\nabla \cdot \vec{\mathbf{P}}$

• $Q_{\text{total}}=\int \rho (r)dV$

Electric dipole

• An electric dipole consists of two equal and opposite charges separated by a distance $d$

Electric dipole moment

• The electric dipole moment $\vec{\mathbf{p}}$ of a dipole is defined as $\vec{\mathbf{p}}=q\vec{\mathbf{d}}$ where $q$ is the magnitude of the charges and $\vec{\mathbf{d}}$ is the vector pointing from the negative charge to the positive charge
  • units: $\mathrm{C}\cdot \mathrm{m}$ (SI unit), and debye ($\mathrm{D}$)
  • (chemistry) molecular dipole moment $\mu$ (often measured in $\mathrm{D}$, and the dipole arrow often points from positive to negative)
• $\vec{\mathbf{\tau }}=\vec{\mathbf{p}}\times \vec{\mathbf{E}}$ is the torque on an electric dipole
• $U=-\vec{\mathbf{p}}\cdot \vec{\mathbf{E}}$ is the potential energy for an electric dipole in an electric field>

Polarization density

• polarization density (or electric polarization, or simply polarization) (קיטוב חשמלי או פּוֹלָרִיזַצְיָה חשמלית)
• $\vec{\mathbf{P}}=\frac{d\vec{\mathbf{p}}}{dV}$ (in $\mathrm{C}\cdot {\mathrm{m}}^{-2}$)
• $\vec{\mathbf{P}}={\epsilon }_0{\chi }_e\vec{\mathbf{E}}$