Electricity

Contents:

  • Charge
  • Electrostatic force
  • Electrical work
  • Capacitance
  • Current
  • Conductance and resistance
  • DC, AC voltage, current and impedance

Electrical charge

  • Physical property of matter
  • Equal charge repels
  • Opposite charge attracts
  • Quantized in elemental charge \(e\).
  • 1\(e\) = \(1.602\times10^{-19}\)
  • Symbol of charge: q, unit: C (Coulomb)
  • Electron charge: -\(e\), proton: \(e\), Neutron: 0
  • Force between charges: \(F = -\frac{k_e q_1 q_1}{r^2} = -\frac{q_1 q_2}{4 \pi \epsilon_0 r^2}\)
  • \(k_e\) = electrical constant, which for vacuum =\(\frac{1}{4\pi \epsilon_0}\)
  • \(\epsilon_0\) = dielectric constant (tabular value)
  • Net charge seeks to external boundaries(surfaces, interfaces)
  • Interior(bulk) phases remain electronneutral
  • Electroneutrality: Sum of negative and positive charges balance (cancel) each other

Electrical field

  • \(X = \frac{F}{Q_{test}}\)
  • Spherical field - inverse square law
  • \(F = -\frac{q_1 q_2}{4 \pi \epsilon_0 r^2}\), \(X = \frac{q_1}{4 \pi \epsilon_0 r^2}\)
  • Fields run from positive to negative (arrows of field towards negative) and shows in what direction a positive charge feels the force.
  • Planar field
  • Charge density of plates \(q\)(C/m\(^2\))
  • Force is independent of distance (\(l\))
  • Total field strength: \(X = \frac{q}{\epsilon}\)

Electrical work

  • Work is force times distance
  • W = \(F\times[-\delta R] = -QX\delta r\)
  • Travel \(\delta r\) is there negative and the work we do is positive.
  • Electric Potential
    • Difference in potential between two sites (constant field):
    • \(U = \phi_B - \phi_A = \frac{W_{A->B}}{Q} = -X\delta r\)
    • 1V = 1J/C (one unit work per unit charge)
    • Potential over variable field(Spherical symmetry):
  • Poisson equation explains how potential varies in space from a net charge.
    • Planar case: areal charge densities \(q_0\) and \(q_L\)
    • \(\frac{d^2\phi}{dx^2} = \frac{-\rho(x)}{\epsilon}\)

Conductance and capacitance

  • Conductors: long range transport

  • Dielectrics(capacitance) only short range transport

    • Insulators

  • Conductors

    • Electronic conductords
      • Metals, semiconds(CB electrons, VB holes)
    • Ionic conductors
      • Anionic, cationic
      • Crystalline: Defects, vacancy, interstitial, intercalation??

  • Capacitance:

  • Stored charge \(Q = \frac{-A\epsilon}{L}\Delta E\)

  • Capacitance \(C = \frac{-Q}{\Delta E} = \frac{\epsilon A}{L}\)

    • Unit: F = C/V
    • \(\epsilon = \epsilon_r \epsilon_0\) Relative*vacuum permitivity.

  • Current:

  • I = Q/t, 1A = 1C/s

  • Current density: i (A/m\(^2\)) = I/A (current devided by arial)

  • Conductivity \(\sigma\) or \(\kappa\) (S/m)=i/X (Ohms Lov)

  • Conductance G

  • Resistitivity \(\rho\)(ohm m)

  • Resistance R (ohm)

Mobility

  • Mechanical mobility B expresses how easy it is to move a species of drift velocity \(v\) by a force F
    • \(v = BF\)
  • Current density is obtained by multiplication with concentration and charge
    • \(i = zecv = zecBF\) (z = # charges, e = elem charge)
  • Charge mobility (or just mobility) is defined as \(u = zeB\)
    • i = zecBF = ucF
  • In electrical field: F = zeX
    • i = ucF=zeucX
  • Conductivity is charge times charge mobility time concentration: \(\sigma = zeuc\)
  • \(i = \sigma X\) (Ohms law)

Exercises

Electricity exercises

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