#4 Slides Notes

Conductors in Electrostatic Equilibrium

Overview of Conductors in Electrostatic Equilibrium

• The electric field inside a conductor in electrostatic equilibrium is zero at all points ($E_{\text{internal}} = 0$).
• If an electric field were present, it would cause charge carriers (such as electrons) to move, disrupting the equilibrium state.
• According to Gauss’s law, the enclosed charge ($Q_{\text{in}}$) within a Gaussian surface inside the conductor is zero:
  • $Q_{\text{in}} = 0$.
• This implies no net charge is enclosed by a Gaussian surface that is located just below the surface of the conductor.

Properties of Electrostatic Conductors

• The external electric field at the surface of the conductor must be perpendicular to that surface.
• Any tangential component ($E<em>{\text{tangential}}$) would exert a force on surface charges, causing them to move and disrupt the electrostatic equilibrium. Therefore, $E</em>{\text{tangential}} = 0$.

Gaussian Surface Analysis

• When a Gaussian surface extends through the conductor's surface:
  • The electric flux only passes through the outer face of the Gaussian surface.
  • The net electric flux ($\Phi_{e}$) through the Gaussian surface can be expressed as:
  • $\Phi<em>{e} = A \times E</em>{\text{surface}}$
  • According to Gauss's law, this is equal to the enclosed charge divided by the permittivity of free space ($\epsilon_{0}$):
  • $\Phi<em>{e} = \frac{Q</em>{\text{in}}}{\epsilon_{0}}$
  • Substituting for enclosed charge ($Q_{\text{in}} = \eta \times A$ where $\eta$ is the surface charge density and $A$ is the area) gives:
  • $A \times E<em>{\text{surface}} = \frac{\eta \times A}{\epsilon</em>{0}}$
  • Hence, the electric field at the surface of the conductor can be deduced:
  • $E<em>{\text{surface}} = \frac{\eta}{\epsilon</em>{0}}$

Electric Field in Charged Conductors with Holes

• Consider a scenario where there is a charged conductor containing a hole.
• The electric field inside the conductor remains zero ($E_{\text{internal}} = 0$).
• Therefore, we can conclude that the enclosed charge on the interior surface of the hole must also be zero:
  • $Q_{\text{in}} = 0$ for the interior surface of the hole.
• Since there’s no electric field inside the conductor, the electric field within the hole must also be zero if there is no charge present.

Effect of a Charge Inside the Hole of a Conductor

• If a charge $q$ is placed inside a hole within a neutral conducting object, it will induce charge on the interior and exterior surfaces of the conductor:
  • A net charge of $-q$ migrates to the inner surface of the conductor, while a net charge of $+q$ remains on the exterior surface.
  • Despite this redistribution of charge, the electric field inside the conductor remains zero ($E_{\text{internal}} = 0$), leading to zero electric flux through any Gaussian surface inside the conductor.

Faraday Cages

Concept of Faraday Cages

• A Faraday cage is a type of conducting box that effectively excludes external electric fields from penetrating a designated region of space.
• The process of utilizing a conducting box to shield against electric fields is known as screening.

Problem-Solving Scenarios

Scenario with Charge in a Hollow Cavity

• Given a charge of +3 nC located in a hollow cavity inside a large, electrically neutral metal chunk, determine the total charge on the exterior surface of the metal:
  • Potential answers:
  • A. 0 nC
  • B. +3 nC
  • C. –3 nC
  • D.