Gauss's Law and Electrostatic Fields

1. Gauss’s Law and Gaussian Surfaces

• Definition: A Gaussian surface is an imaginary 3-dimensional closed surface used to apply Gauss’s Law, chosen based on the geometry of the charges being studied.
• Core Principle: The law relates the electric flux through a surface to the charge enclosed within that surface.

2. Electric Fields in Spherical Systems

• Hollow Charged Sphere:
  • Inside ($r < a$): The electric field is zero because there is no enclosed charge; all charge is evenly distributed on the surface.
  • Outside ($r > a$): The field behaves as if all the charge is concentrated at a point at the centre of the sphere: $E = \frac{Q}{4\pi\epsilon_0 r^2}$.
• Uniformly Charged Solid (Non-conducting) Sphere:
  • Inside: The field increases linearly with the radial distance from the centre ($r$): $E = \frac{Qr}{4\pi\epsilon_0 a^3}$.
  • Outside: The field remains the same as that of a hollow sphere, decreasing with the square of the distance ($1/r^2$).
• Conducting Spheres (Hollow or Solid):
  • In a conductor, charges move to the surface to cancel out the internal field.
  • Therefore, the electric field inside a conducting sphere is always zero, regardless of whether it is solid or hollow.

3. Electric Potential in Spherical Systems

• Inside a Hollow Sphere: Because the electric field is zero ($dV/dr = 0$), the electric potential ($V$) is constant.
• Outside a Hollow Sphere: The potential is calculated as $V = \frac{Q}{4\pi\epsilon_0 r}$.
• Equipotential Surfaces: For a point charge or sphere, these are concentric circles where the radius is constant, and the electric field is always perpendicular to these surfaces.

4. Fields of Planes, Sheets, and Capacitors

• Charged Sheet: A sheet has two sets of field lines (up and down), resulting in a uniform electric field of $E = \frac{\sigma}{2\epsilon_0}$.
• Charged Conducting Plane: If there is only one set of field lines, the field is $E = \frac{\sigma}{\epsilon_0}$.
• Parallel Plate Capacitor:
  • The fields from the two plates reinforce each other inside the capacitor and cancel out outside.
  • The field inside is uniform: $E = \frac{\sigma}{\epsilon_0}$.
  • Capacitance ($C$): For parallel plates, it is defined as $C = \frac{\epsilon_0 A}{d}$.

5. Dynamics of a Charged Particle in an Electric Field

• Force and Acceleration: A charge ($q$) of mass ($m$) in a uniform electric field ($E$) experiences a constant force $F = qE$ and a constant acceleration $a = \frac{qE}{m}$.
• Projectile Motion: When launched into a parallel plate capacitor, the particle follows a projectile path.
  • Motion in the x-direction is uniform ($x = ut$).
  • Motion in the y-direction is accelerated ($y = \frac{1}{2}at^2$).
• Example calculation: For a $3\mu C$ charge with a mass of $10^{-8}$ kg moving at $2 m/s$, its position after 1 second would be $(2, 1.5 \times 10^2)$ metres, assuming specific field parameters provided in the text.