PHYC10160 Th2.1 Electricity 1: Notes on Charge, Force, Field, Gauss's Law, and Conductors

  • Overview of electricity topics covered: charge, force, field, Coulomb's law, electric field from point charges, superposition, Gauss's law and electric flux, and electrostatic conductors (including cavities, surface charges, and field behavior).
  • Atom basics and charges: electrons (-e), protons (+e), neutrons (0); atomic number Z (number of protons/electrons); mass number A; atoms are neutral overall; elementary charge e = 1.602×10^-19 C.
  • Charge conservation: total charge is conserved in all processes, including nuclear reactions and rubbing/contact charge transfer.
  • Conductors vs insulators vs semiconductors: free conduction electrons in conductors; insulators with bound charges; semiconductors have conductivity that can be tuned (doping, fields).
  • Vector form of Coulomb forces and superposition: forces are vector quantities; total force is the vector sum of contributions from all charges.
  • Electric field concept: E = F/q0; for a point charge q, E = (1/(4π ε0)) q / r^2 in the radial direction; field lines and symmetry help visualize E.
  • Dipoles and a simple dipole field: dipole moment p = q d (from negative to positive charge); field of a dipole given by E(r) = (1/(4π ε0)) [ (3(p·r̂) r̂ − p) / r^3 ]; along the dipole axis, E = (1/(4π ε0)) (2p) / r^3.
  • Gauss's law and flux: ΦE = ∮ E · dA = qin/ε0; flux through a closed surface depends only on enclosed charge; symmetry helps choose Gaussian surfaces.
  • Field lines: locally tangent to E; density of lines indicates field magnitude; lines are perpendicular to conductors’ surfaces just outside.
  • Conductors in electrostatic equilibrium: E_inside = 0; charges reside on the surface; outside field is perpendicular to surface and has magnitude σ/ε0; irregular conductors have higher surface charge density where curvature is smallest.
  • Cavities in conductors: if no charges are inside a cavity, the cavity walls carry no net charge; all excess charge resides on the outer surface; if charges are placed inside a cavity, induced charges appear on the cavity walls to ensure E_inside = 0 in the conductor.
  • Infinite charged sheets: for a nonconducting sheet with surface charge density σ, the field on each side is E = σ/(2ε0) and points away from the sheet for σ > 0; translational symmetry applies.
  • Planes and spheres: infinite plane fields are uniform and perpendicular; spherical symmetry leads to E outside a charged spherical shell acting like a point charge at the center and E inside vanishing for a hollow shell or a solid conductor in electrostatic equilibrium.
  • Dipole relevance to real systems: dipole interactions and energy transfer (e.g., photosynthesis) illustrate how molecular-scale dipoles affect fields and energy transport.
  • All numerical references included: masses, radii, charges, constants, and derived formulas are presented in standard SI units; ε0 is the vacuum permittivity.
  • Key equations to remember (LaTeX):
    • Coulomb's law: F<em>12=14πε</em>0q<em>1q</em>2r2r^ .\mathbf{F}<em>{12}=\frac{1}{4\pi\varepsilon</em>0}\frac{q<em>1 q</em>2}{r^2}\hat{\mathbf{r}}\,.
    • Electric field from a point charge: E=14πε0qr2r^ .\mathbf{E}=\frac{1}{4\pi\varepsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}\,.
    • Superposition (net field): E=∑<em>iE</em>i=14πε<em>0∑</em>iq<em>ir−r</em>i∣r−ri∣3 .\mathbf{E}=\sum<em>i \mathbf{E}</em>i = \frac{1}{4\pi\varepsilon<em>0}\sum</em>i q<em>i \frac{\mathbf{r}-\mathbf{r}</em>i}{|\mathbf{r}-\mathbf{r}_i|^3}\,.
    • Electric flux through a surface: ΦE=∮E⋅dA .\Phi_E=\oint \mathbf{E}\cdot d\mathbf{A}\,.
    • Gauss's law: ∮E⋅dA=q<em>inε</em>0 .\oint \mathbf{E}\cdot d\mathbf{A}=\frac{q<em>{\text{in}}}{\varepsilon</em>0}\,.
    • Dipole moment: p=q d  (direction from -q to +q) .\mathbf{p}=q\,\mathbf{d}\; (\text{direction from -q to +q})\,.
    • Field of a dipole: E(r)=14πε<em>01r3(3(p⋅r^)r^−p) .\mathbf{E}(\mathbf{r})=\frac{1}{4\pi\varepsilon<em>0}\frac{1}{r^3}\Bigl(3(\mathbf{p}\cdot \hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{p}\Bigr)\,. (Along the axis, this reduces to E=14πε</em>02pr3\mathbf{E}=\frac{1}{4\pi\varepsilon</em>0}\frac{2\mathbf{p}}{r^3}.)
  • Notation: Z = atomic number (protons/electrons), A = mass number (protons + neutrons). Example: Uranium-235 has Z = 92 and A = 235.
  • Surface charge density: σ=qA  .\sigma=\frac{q}{A}\;.
  • For an infinite uniformly charged sheet (nonconducting): E=±σ2ε0n^ .\mathbf{E}=\pm \frac{\sigma}{2\varepsilon_0}\hat{\mathbf{n}}\,.