Electromagnetism Notes: Fundamentals, Charge, and Line-Charge Field

Electric charges, atoms, and the foundations

• Four Fundamental Forces (mentioned on the slide):
  • Weak (nuclear) force
  • Strong force (holds the nucleus together)
  • Electromagnetic force
  • Gravity (often included as the fourth fundamental force; not explicitly listed due to transcription, but typically part of the four)
• Atoms and charges
  • Conductors: Electrons are free to move; good conductors have a macroscopic conduction band.
  • Insulators (dielectrics): Charge tends to stay put but can be polarized by external fields.
  • Neutral atom: Equal numbers of positive (protons) and negative (electrons) charges → electrical neutrality.
  • Conservation of charge: Charge cannot be created or destroyed (global conservation).
• Fundamental constants and elementary charge
  • Elementary charge: $e = 1.602176634 \times 10^{-19}\ \mathrm{C} \approx 1.6 \times 10^{-19}\ \mathrm{C}$
  • Permittivity of vacuum: $\varepsilon_0 \approx 8.854187817 \times 10^{-12}\ \mathrm{F/m}$
  • Coulomb constant: $k<em>e = \frac{1}{4\pi \varepsilon</em>0} \approx 8.98755 \times 10^{9}\ \mathrm{N\,m^2/C^2}$

Coulomb's Law and the electric field

• Coulomb's Law (magnitude):
  • $F = k<em>e \frac{|q</em>1 q_2|}{r^2}$
  • Direction: along the line joining the charges; like charges repel, opposite charges attract.
• Electric field (definition and point-charge form)
  • Electric field as force per unit charge: $\mathbf{E} = \frac{\mathbf{F}}{q}$
  • For a point charge: $\mathbf{E}(\mathbf{r}) = k_e \frac{q}{r^2}\ \hat{\mathbf{r}}$
• Relation between force, field, and test charge
  • For a test charge $q<em>0$ placed at a point with field $\mathbf{E}$: $\mathbf{F} = q</em>0 \mathbf{E}$

Electric field concepts with a line of charge (finite line)

• Problem setup: Find the force on a test charge $q$ due to a very thin line with uniform linear charge density $\lambda = \dfrac{dq}{dx}$, where the test charge is located a distance $a$ from the closer end of the line (line of length $L$ along the x-axis from $x=0$ to $x=L$).
• Elemental charge and distance
  • A differential element: $dq = \lambda \, dx$
  • Distance from the element at position $x$ to the test charge: $r = \sqrt{x^2 + a^2}$
• Differential electric field from a small element
  • Magnitude: $dE = k<em>e \frac{dq}{r^2} = k</em>e \frac{\lambda \ dx}{x^2 + a^2}$
  • Angle between the field vector and the perpendicular direction: $\cos\theta = \frac{a}{r} = \frac{a}{\sqrt{x^2 + a^2}}$
  • Component of the field perpendicular to the line (along the y-direction):
    $dE<em>y = dE \cos\theta = k</em>e \lambda a \frac{dx}{(x^2 + a^2)^{3/2}}$
• Integrating over the line (0 to L)
  • Total electric field in the y-direction:
    $E<em>y = k</em>e \lambda a \int_{0}^{L} \frac{dx}{(x^2 + a^2)^{3/2}}$
  • Evaluate the integral:
    $\int \frac{dx}{(x^2 + a^2)^{3/2}} = \frac{x}{a^2 \sqrt{x^2 + a^2}} + C$
  • Therefore,
    $E<em>y = k</em>e \lambda a \left[ \frac{x}{a^2 \sqrt{x^2 + a^2}} \right]<em>{0}^{L} = k</em>e \lambda \frac{L}{a \sqrt{L^2 + a^2}}$
• Result for a finite line
  • Magnitude and direction: $\mathbf{E} = E_y \, \hat{\mathbf{y}}$ with
  • $E<em>y = k</em>e \lambda \frac{L}{a \sqrt{L^2 + a^2}}$
  • Force on the test charge: $\mathbf{F} = q \mathbf{E} = q \; E_y \, \hat{\mathbf{y}}$
• Limiting cases
  • Semi-infinite line ($L \to \infty$):
  • $E<em>y \to \frac{k</em>e \lambda}{a}$
  • Infinite line (line extends both directions, from $-\infty$ to $+\infty$):
  • The field is doubled (symmetric from both sides):
  • $E<em>{\text{infinite line}} = \frac{2 k</em>e \lambda}{a} = \frac{\lambda}{2 \pi \varepsilon_0 a}$
• Notes on direction
  • For positive line charge and a positive test charge, the direction of the field is away from the line in the perpendicular direction considered; the sign of the charge determines the direction of the force accordingly.

Practical and conceptual recap

• Conduction vs insulation in materials
  • Conductors: delocalized electrons enable current flow; charges rearrange to shield internal regions from external fields (electrostatic shielding).
  • Dielectrics: charges are bound, but the material can be polarized by external fields, producing induced dipoles that modify the field inside the material.
• Charge conservation and neutrality underpin how charges distribute in materials and how external fields influence them.
• The electron charge and Coulomb constant are foundational for calculating forces and fields at the microscopic scale; these feed into macroscopic laws (Gauss’s law, etc.) when extended to symmetry shapes and charge distributions.
• The line-charge calculation demonstrates converting a distributed source into an integral expression for the resultant field, illustrating the connection between microscopic charge elements and macroscopic fields. It also highlights how geometry (finite length, semi-infinite, infinite) changes the resulting field and force.

Key formulas to memorize

• Coulomb's Law (point charges):
  $\mathbf{F} = k<em>e \frac{q</em>1 q<em>2}{r^2} \hat{\mathbf{r}}, \quad k</em>e = \frac{1}{4\pi \varepsilon_0} \approx 8.98755 \times 10^{9}\ \mathrm{N\,m^2/C^2}$
• Electric field of a point charge:
  $\mathbf{E}(\mathbf{r}) = k_e \frac{q}{r^2} \hat{\mathbf{r}}$
• Electric field as force per unit charge:
  $\mathbf{E} = \frac{\mathbf{F}}{q}$
• Elementary charge:
  $e = 1.602176634 \times 10^{-19}\ \mathrm{C}$
• Line charge field (finite line, perpendicular distance a, length L):
  $E<em>y = k</em>e \lambda \frac{L}{a \sqrt{L^2 + a^2}}$
  $F<em>y = q E</em>y$
• Infinite line field (Gauss’s law check):
  $E = \frac{\lambda}{2 \pi \varepsilon<em>0 r} = \frac{2 k</em>e \lambda}{r}$

If you want, I can add more worked steps for the line-charge integral or generate a quick cheat-sheet with the essential constants and common limiting cases for quick study.