Uniformly Charged Rod

Electric Flux Through Coaxial Cylindrical Surfaces

When a cylindrical rod of radius r0 has a uniform volume charge density ρ and is aligned with the x-axis, the electric flux through different coaxial cylindrical Gaussian surfaces can be analyzed using Gauss's law.

• Gauss's Law: The total electric flux Φe through a closed surface is given by:

• Flux Calculation for Two Cases: For a cylinder of length 2l and radius r/2: For a cylinder of length 2l and radius r: Ratio of fluxes:

• Example: If the charge density and dimensions are known, the flux through the larger Gaussian surface is twice that through the smaller one.

Electric Field and Electric Flux

Properties of Electric Flux in Cylindrical Geometry

For a cylindrical rod with uniform charge density, the electric flux through different parts of a coaxial Gaussian surface can be determined by symmetry and field direction.

• Flux through Flat End Surfaces: The electric field is parallel to these surfaces, so the flux is zero.

• Flux through Curved Surface: The electric field is perpendicular to the curved surface, and its magnitude is constant at a fixed radius.

• Key Points:

  • Flux through the flat end surfaces is zero.

  • Magnitude of the electric field is constant over the curved surface.

• Example: For a Gaussian cylinder surrounding the rod, only the curved surface contributes to the total flux.

Electric Field Outside a Cylinder

Field Behavior for r > r0

Outside a uniformly charged cylinder, the electric field can be found using Gauss's law and symmetry arguments.

• Gauss's Law Application:

• Field on Curved Surface:

• Key Point: The magnitude of the electric field outside the cylinder is inversely proportional to r.

• Example: At a distance r from the axis, the field decreases as r increases.

Electric Field Inside a Cylinder

Field Behavior for r < r0

Inside a uniformly charged cylinder, the electric field increases linearly with distance from the axis.

• Gauss's Law Application:

• Field on Curved Surface:

• Key Point: The magnitude of the electric field inside the cylinder is directly proportional to r.

• Example: At the center (r = 0), the field is zero; it increases linearly as you move outward.

Gaussian Surface Choice

Symmetry and Calculating Electric Field

Choosing an appropriate Gaussian surface is crucial for applying Gauss's law effectively. The surface must exploit the symmetry of the charge distribution.

• Symmetry Requirement: The electric field must be constant in magnitude and direction over the surface for Gauss's law to simplify calculations.

• Example: For a cubical shell of charge, neither a cube nor a sphere centered on the origin allows calculation of the field at point P using only Gauss's law, due to lack of symmetry.

• Key Point: If the field at point P is not equal to the field at all points on the surface, Gauss's law cannot be used directly to find the field at P.

Conductors in Electrostatic Equilibrium

Properties of Conductors at Equilibrium

In electrostatic equilibrium, conductors exhibit specific properties regarding charge distribution and electric field.

• Key Properties:

  • All charges are stationary.

  • The electric field inside the conductor is zero.

  • The electric field at the surface is perpendicular to the surface.

  • All excess charge resides on the surface of the conductor.

• Example: A conductor with an irregular shape will have all excess charge distributed on its outer surface.

Charge Distribution in a Metal Bucket

Electrostatic Equilibrium with Internal Charge

When a metal bucket with zero excess charge contains a metal ball with charge +q, the distribution of charge on the bucket's surfaces can be determined using Gauss's law and the properties of conductors.

• Inner Surface: The charge on the inner surface of the bucket is –q to ensure the electric field inside the metal is zero.

• Outer Surface: The charge on the outer surface of the bucket remains zero, since no net charge has been added to the bucket itself.

• Key Point: The induced charge on the inner surface exactly cancels the charge of the ball, maintaining electrostatic equilibrium.

• Example: If a +2 μC ball is placed inside, the inner surface acquires –2 μC, and the outer surface remains neutral.

Additional info: These notes expand on the brief points in the slides, providing full academic context and explanations suitable for college-level physics students.