Lecture 5

Charge Density

• Overview: Fundamental concept in understanding Electric Fields due to charge distributions.

Electric Field of a Charge Distribution

• The electric field produced by a distribution of charge is calculated by considering the contributions from each infinitesimal amount of charge.

Electric Field of a Charged Rod

• Focus on how the geometry of a charged rod impacts the electric field at different points around it.

Insulators and Conductors

Insulators:

• Characteristics:

  • No mobile charges; all electrons are bound to their own atoms (examples: glass, wood, plastic).

  • Excess charge remains stationary on the surface or within the insulator since electrons cannot move freely.

Conductors:

• Key Properties:

  • Have mobile charges which enable current flow (examples: metals, ionic solutions).

  • Charge distribution occurs on the surface when in electrostatic equilibrium, leading to a net electric field (E) of zero within the conductor.

Polarization and Charge Redistribution

• Care must be taken not to overgeneralize; the net electric field inside a conductor is only zero under static conditions or equilibrium.

• Dynamic conditions, such as charging or during electric circuits, can create non-zero electric fields.

Discharging by Contact and Grounding

• Discharging Mechanism:

  • When a charged object approaches a body, it causes a polarization of charges (e.g., sweat, blood). Upon contact, charges redistribute over a larger surface.

  • Grounding involves connecting to the earth, allowing charges to flow freely; for instance, a negatively charged ball discharging via a grounded body.

Induction Charging of a Conductor

• Process:

  1. Bring a charged rod close to the conductor.

  2. Ground the conductor.

  3. Remove the ground connection before taking away the charged rod, resulting in a charged conductor.

Electric Field Characterization with iClicker Questions

Question on Two Metal Balls:

• Two uncharged metal balls and a charged ball (Z) are manipulated using connecting wires.

  • The manipulation leads to a state where the balls become charged in opposite manners.

Charge Distribution in Conductors with Cavities:

• Analyzing the influence of point charges inside the cavity on the inner and outer surface charges on the conductor.

• Internal charges adjust to maintain overall charge balance and shielding.

Superposition Principle

• Concept:

  • The net electric field at a point in space is the vector sum of fields from individual point charges.

  • Charge distributions can be viewed as collections of point charges, leading to calculations via integration of their electric fields.

Steps for Electric Field Calculations from Charged Distributions

1. Geometry Understanding: Analyze the shape of the charge distribution.

2. Select Charge Element: Define dq (the infinitesimal charge element).

3. Evaluate dE Contribution: Determine the electric field contribution from dq.

4. Exploit Symmetry: Use symmetry where applicable to simplify calculations.

5. Set Up Integral: Formulate the integral for total charge distribution.

6. Solve Integral: Perform the integration.

7. Result Verification: Ensure results make sense and are in the correct units.

Electric Field in the Bisecting Plane of a Charged Rod

• Calculating electric fields involves considering geometry and symmetry.

• Key Formula: E = ( \frac{1}{4\pi \epsilon_0} \frac{Q}{r^2} \hat{r} ) represents the electric field due to point charges, adjusted for rod geometry.

Behavior of Electric Fields for Finite and Infinite Rods

• Infinite Rod: Electric fields are defined for all space due to uniform charge distribution.

• Finite Rod: Fields are analyzed on the bisecting plane, demonstrating localized effects, transitioning towards point charge behavior at large distances.

General Procedure for Calculating Electric Fields from Distributed Charges

1. Segment the distribution to manageable pieces.

2. Write electric field expressions for single charge pieces and their components.

3. Integrate the contributions accordingly to find the overall field.

4. Validate the resultant physics metrics: direction, units, and expected behavior in limiting cases.