2.3 Electric fields of point charges and continuous charge distributions

Electric fields are crucial in understanding how charges interact. This section dives into the fields created by point charges and continuous charge distributions, showing how to calculate their strengths and directions.

Gauss's law is introduced as a powerful tool for finding electric fields in symmetric situations. It connects the electric flux through a closed surface to the enclosed charge, simplifying calculations for various charge distributions.

Electric Fields of Point Charges

Point Charge Field and Dipole Field

• Point charge field represents the electric field generated by a single point charge
  • Calculated using Coulomb's law: $E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}$, where $q$ is the charge and $r$ is the distance from the charge
  • Field lines radiate outward for positive charges and inward for negative charges (proton and electron)
• Dipole field is the electric field created by two equal and opposite point charges separated by a small distance
  • Field lines start from the positive charge and end on the negative charge
  • Field strength decreases rapidly with distance from the dipole (water molecule)

Charge Density and Symmetry Considerations

• Charge density ($\rho$) is the amount of electric charge per unit volume, area, or length
  • Volume charge density: $\rho = \frac{dq}{dV}$, where $dq$ is the charge in a small volume $dV$
  • Surface charge density: $\sigma = \frac{dq}{dA}$, where $dq$ is the charge on a small area $dA$
  • Linear charge density: $\lambda = \frac{dq}{dl}$, where $dq$ is the charge on a small length $dl$
• Symmetry considerations simplify the calculation of electric fields
  • Spherical symmetry: field depends only on the distance from the center (charged metal sphere)
  • Cylindrical symmetry: field depends on the distance from the axis and the angle (charged wire)
  • Planar symmetry: field is constant in magnitude and direction (charged infinite plane)

Continuous Charge Distributions

Line, Surface, and Volume Charge Distributions

• Line charge is a continuous distribution of electric charge along a line or curve
  • Electric field is calculated using the linear charge density $\lambda$ (charged rod)
• Surface charge is a continuous distribution of electric charge on a surface
  • Electric field is calculated using the surface charge density $\sigma$ (charged metal plate)
• Volume charge is a continuous distribution of electric charge throughout a volume
  • Electric field is calculated using the volume charge density $\rho$ (charged dielectric material)

Calculating Electric Fields from Charge Distributions

• Electric field due to a continuous charge distribution is found by integrating the contributions from each infinitesimal charge element
  • Line charge: $dE = \frac{1}{4\pi\epsilon_0}\frac{dq}{r^2} = \frac{1}{4\pi\epsilon_0}\frac{\lambda dl}{r^2}$
  • Surface charge: $dE = \frac{1}{4\pi\epsilon_0}\frac{dq}{r^2} = \frac{1}{4\pi\epsilon_0}\frac{\sigma dA}{r^2}$
  • Volume charge: $dE = \frac{1}{4\pi\epsilon_0}\frac{dq}{r^2} = \frac{1}{4\pi\epsilon_0}\frac{\rho dV}{r^2}$
• The total electric field is the vector sum of the contributions from all charge elements (principle of superposition)

Gauss's Law

Gauss's Law and Its Applications

• Gauss's law relates the total electric flux through a closed surface to the net charge enclosed by the surface
  • Mathematically expressed as: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$, where $\vec{E}$ is the electric field, $d\vec{A}$ is an infinitesimal area element, and $Q_{enc}$ is the net charge enclosed
• Gauss's law is particularly useful for calculating electric fields in situations with high symmetry
  • Spherical symmetry: $E = \frac{1}{4\pi\epsilon_0}\frac{Q_{enc}}{r^2}$ (charged spherical shell)
  • Cylindrical symmetry: $E = \frac{1}{2\pi\epsilon_0}\frac{\lambda}{r}$ (infinitely long charged wire)
  • Planar symmetry: $E = \frac{\sigma}{2\epsilon_0}$ (uniformly charged infinite plane)

Charge Density and Point Charge Field in Gauss's Law

• Charge density appears in the integral form of Gauss's law when dealing with continuous charge distributions
  • $\oint \vec{E} \cdot d\vec{A} = \frac{1}{\epsilon_0} \int \rho dV$ for volume charge density
  • $\oint \vec{E} \cdot d\vec{A} = \frac{1}{\epsilon_0} \int \sigma dA$ for surface charge density
  • $\oint \vec{E} \cdot d\vec{A} = \frac{1}{\epsilon_0} \int \lambda dl$ for linear charge density
• Point charge field can be derived from Gauss's law by considering a spherical surface enclosing the point charge
  • Applying Gauss's law: $\oint \vec{E} \cdot d\vec{A} = \frac{q}{\epsilon_0}$
  • Due to spherical symmetry, $E$ is constant on the surface and perpendicular to it, so $\oint \vec{E} \cdot d\vec{A} = E \oint dA = E(4\pi r^2)$
  • Equating the two expressions yields the point charge field: $E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}$