9/22 Lecture Notes

Recap of Previous Class

• Exam Discussion: The previous class covered details about the upcoming exam.
• Finding Electric Field using Gaussian's Law: We discussed applying Gaussian's Law in several scenarios:
  • Long charged wire or cable.
  • Plane of charge.
  • Sphere of charge (introduced in the last class).
  • Slab of charge (also introduced).

Gaussian's Law

• Requirement for Gaussian's Law: To find the electric field for any charge distribution (long wire, plane, sphere, slab), we always need to consider an imaginary closed surface called a Gaussian surface.
• Gaussian's Law Definition: The total electric flux ($\Phi<em>E$) enclosed by a closed Gaussian surface is equal to the total charge enclosed ($Q</em>{enc}$) within that surface divided by the permittivity of free space ($\epsilon_0$).
  • Mathematically: $\Phi<em>E = \oint \vec{E} \cdot d\vec{A} = \frac{Q</em>{enc}}{\epsilon_0}$
  • Here, $\vec{E}$ is the electric field and $d\vec{A}$ is an infinitesimal area vector on the Gaussian surface.

Conductors in Electrostatic Equilibrium

• Charge Distribution: If extra charge is added to a conductor, it always resides on the surface of the conductor.
  • Reasoning: Like charges repel each other. To maximize their distance and minimize their electrostatic potential energy, the excess charges push away from each other and settle on the outermost surface of the conductor. They reach an equilibrium state where they are as far apart as possible.
• Electric Field Inside a Conductor: In electrostatic equilibrium, the electric field inside a conductor is zero ($E_{inside} = 0$).
  • Implication: Since there is no charge inside and the electric field is zero, any net charge resides solely on the surface.

Applications and Examples of Conductors

• Safety in Airplanes during Lightning: People inside an airplane are safe during a lightning strike because the airplane acts as a conductor. The lightning charge accumulates on the outer surface of the airplane, but the electric field inside the conducting structure remains zero, protecting the occupants.
• Lightning Rods: Tall buildings are equipped with lightning rods. If lightning strikes or charge accumulates from clouds, the charge flows onto the surface of the building and is safely guided through the lightning rod to the Earth (which acts as an infinite source/sink of charge). This prevents damage to the building's interior, as the electric field inside remains zero.
• Performers with Metal Suits: A performer wearing a metal suit can be subjected to high voltage, causing visible electrical discharges around their body, yet remain unharmed. This is because the metal suit acts as a conductor, and the electric field inside (where the performer is) is zero.
• Conductor with a Hole: If a conductor has a hole, any excess charge still sits on the outer surface of the conductor, not within the hole itself unless there's an isolated charge placed inside the hole.

Faraday Cage (Shielding/Screening)

• Purpose: A Faraday cage is a conducting enclosure designed to block external electric fields from penetrating its interior, creating a region of zero electric field inside.
• Mechanism: When a conducting box (Faraday cage) is placed within an external electric field, the free charges within the conductor redistribute themselves in a way that creates an internal electric field that exactly cancels out the external field within the enclosure. As a result, the net electric field inside the conducting box is zero.
• Real-world Relevance: This concept is crucial in designing sensitive electronic devices where manipulating or blocking electric fields is necessary (e.g., semiconductor manufacturing, shielding against electromagnetic interference).

Electric Field Inside and Outside a Slab of Charge (Gaussian's Law Example)

• Scenario: Consider a charged slab situated on the $xy$-plane, with charge changing in the $z$-direction (e.g., from $-a$ to $+a$). We want to find the electric field inside and outside.
• Finding Electric Field Inside the Slab:
  • Gaussian Surface: To find the electric field inside the slab, we construct a cylindrical (or pillbox) Gaussian surface inside the slab, with its flat ends parallel to the $xy$-plane and an infinitesimal thickness $dz$ along the $z$-axis. The area of the ends is $A$.
  • Charge Enclosed ($Q_{enc}$): For a small region of thickness $dz$ centered at $z$ inside the slab, the charge enclosed is given by the volume charge density ($\rho$) multiplied by the volume of the Gaussian surface (Area $A$ times thickness $dz$).
    • If assuming uniform charge density for this small region, $\Delta Q_{enc} = \rho \cdot A \cdot \Delta z$
    • To find the total charge enclosed up to a certain $z$ from the center, we integrate: $Q<em>{enc}(z) = \int</em>0^z \rho(z') A dz'$
    • If the charge distribution is symmetrical around the $xy$-plane, one might consider the charge enclosed from $-z$ to $+z$ for a Gaussian surface spanning that range, or $2 \times \int_0^z \rho(z') A dz'$.
  • Applying Gauss's Law: $\oint \vec{E} \cdot d\vec{A} = E<em>{inside} \cdot 2A = \frac{Q</em>{enc}(z)}{\epsilon_0}$ (assuming $E$ is uniform across the ends and perpendicular to them, and zero flux through the side walls).
  • Result: The electric field inside ($E<em>{inside}$) will be a function of $z$, as $Q</em>{enc}$ changes with $z$.
• Finding Electric Field Outside the Slab:
  • Gaussian Surface: Construct a Gaussian surface outside the slab, again a cylinder or pillbox, with its ends parallel to the $xy$-plane, where one end is outside the slab (e.g., at z > a) and the other is either inside the slab (at $z=0$ or $-a$) or outside on the other side. Typically, it encloses the entire slab.
  • Charge Enclosed ($Q<em>{enc}$): When the Gaussian surface is outside the slab (e.g., for z > a), the total charge enclosed is the charge of the entire slab. This means the integration for $Q</em>{enc}$ would be from $-a$ to $+a$ (or $2 \times \int_0^a \rho(z') A dz'$).
  • Result: The electric field outside ($E_{outside}$) will be constant with respect to $z$ once entirely outside the slab, as the total enclosed charge is constant.

Electric Potential and Electric Potential Energy

• Electric Potential ($V$):
  • Other Names: Often called potential difference or voltage.
  • Definition: Electric potential is always measured in terms of a potential difference between two points.
  • Reference Point: If