E3 in class review

Magnetic Field Calculation
- Formula: Magnetic field ( ext{B}) is equal to ( \mu_0 I - 2 \psi r )
- Constants in use: ( \mu_0 = 4\pi \times 10^{-7} \text{ T m/A} ), current ( I = 2 ext{ amps} )， radius considerations are crucial.
Frequency Consideration
- Recognize the significance of frequency in context; relates to the constants in use.

Frequently arises questions about the radius used in calculations.
Specified radius for computing the magnetic field at 10 centimeters is ( r = 0.1 ext{ m} ).
Ensures clarity to avoid confusion regarding which radius to apply in similar problems.

Magnitude Inside a Capacitor Plate
  - Examining magnetic field at point inside the capacitor plate (given radius ( r = 2 ext{ cm} ) or ( 0.02 ext{ m} )).
  - Use of Ampère's Law: ( \oint \mathbf{B} \cdot d ext{l} = \mu_0 I_{enc} )
  - Focus on the enclosed current within smaller radius of interest.
Enclosed Displacement Current Calculation
  - Enclosed displacement current is not the full 2 amps; must consider proportionality with area.
  - Ratio of areas:
   - Total displacement: ( I/A ) is constant.
  - For area covered by smaller circle: (\text{Area} = \pi (0.02)^2 ) and total area ( \pi (0.05)^2 )
  - Resulting in ( I_{enclosed} \approx 0.32 ext{ amps} ).
Displacement Current and Magnetic Field Relation
- Magnetic field relationship: ( E = \frac{\mu_0 I_{enclosed}}{2 \pi r} ).
- Applying specific values, ( I_{enclosed} = 0.32 ext{ amps} ) and ( r = 0.02 ext{ m} ).

Discussion on relating displacement current and changing electric field, ( \frac{d \Phi_E}{dt} ).
Utilize the equation: ( I_d = \epsilon_0 \frac{d \Phi_E}{dt} )
- ( \Phi_E = \int E \, dA = E A ).
- For a static capacitor, area is constant enabling simplified calculations.
Discussion about the possibility of assessing displacement currents in discharging cases, with directionality of the flow reversed.

Charging Capacitor Magnetic Field Direction
  - When illustrating directionality with right-hand rule:
    - Thumb indicates current direction.
    - Fingers denote magnetic field direction.
  - Comparison between capacitor electric field directions indicates behavioral similarities.

Exam Structure and Material
  - Coverage of topics spanning chapters 28, 29, 30, and 32.
  - Exclusions noted for Chapter 31.
  - Important for students to have clarity on exam content and associated lecture material.

Ensure to get sufficient sleep before the exam for optimal cognitive function.
Emphasis on showing complete work for problems to avoid scoring penalties.
Precision in writing answers, especially for graphing, labeling axes correctly.
- Units must be included in answers to uphold clarity and prevent point deductions.

Magnetic Forces Equations:
  - For charged particles: ( \mathbf{F} = q (\mathbf{v} \times \mathbf{B}) ).
  - For current-carrying wires: ( \mathbf{F} = I (\mathbf{L} \times \mathbf{B}) ).
  - Right-hand rule utilized for determining directions of forces upon conditions present in magnetic fields.
Ampère's Law Discussion:
- Integrated form ( \oint \mathbf{B} \cdot d ext{l} = \mu_0 I_{enc} ).
- Clarifications made around assessment of magnetic field inside wires/capacitors with treatment of displacement current.

Situation Context:
- Change in magnetic flux through a loop induces EMF, covered by Lenz's Law — producing opposing effects to original flux changes.
Computational Approach:
- Recognize how induced current operates; factor in changing variables: ( \Phi_B = ext{B} \cdot A \cdot cos(\theta) ).
- Must attend to how induced currents relate directly back to changes in original flux.
Momentum Handling of Induced Fields:
- Employ right-hand rule adjustments for direct analysis on induction phenomena or altered magnetic flux regimes across circuits.

Overview of the four Maxwell's equations relation to electricity and magnetism.
Additional focus on Gauss's law for magnetism and the context of displacement currents in electromagnetic fields.
Preparation Recommendation:
- All equations and principles should be memorized.
- Special attention to right-hand rules and applying them efficiently during problem-solving scenarios.