pages/30_PHY_613_Electromagnetic_Theory_II.py

import streamlit as st

# Set the page title
st.title("PHY 613: Electromagnetic Theory II")

# Course Details
st.markdown("""
## Course Details
- **Course Title**: Electromagnetic Theory II  
- **Credits**: 3  
- **Prerequisites**: PHY 611  
- **Instructor**: [Instructor Name]  
- **Office Hours**: [Office Hours]  

## Course Description
This course builds upon Electromagnetic Theory I and focuses on advanced topics in electromagnetism, including electromagnetic wave propagation, optical phenomena, radiation theory, and the covariant formulation of Maxwell’s equations.

## Course Objectives
By the end of this course, students will:
- Be proficient in the theory of electromagnetic waves.
- Understand the interaction of electromagnetic radiation with matter.
- Apply the covariant formulation of Maxwell’s equations to physical problems.

---
""")

# Weekly Outline with Problems

# Week 1: Review of Maxwell’s Equations
with st.expander("**Week 1: Review of Maxwell’s Equations**"):
    st.markdown("""
    ### Topics Covered
    - Maxwell’s Equations: Review of integral and differential forms of Maxwell’s equations in free space and in materials.
    - Energy and Momentum Conservation: Derivation and applications of the Poynting theorem.
    - Boundary Conditions: Conditions on electric and magnetic fields at material interfaces.
    
    ### Problems
    1. Derive Maxwell’s equations in a vacuum from the differential forms of Gauss’s law and Ampère’s law with displacement current.
    2. Solve for the electric and magnetic fields in a region with a time-varying current distribution.
    3. Calculate the energy stored in the electric field of a spherical charge distribution.
    4. Derive the Poynting vector from Maxwell’s equations and compute the energy flux through a closed surface.
    5. Apply Maxwell’s equations to solve for the fields around a current loop in free space.
    6. Solve for the boundary conditions of the electric and magnetic fields at the interface between two dielectrics.
    """)

# Week 2-3: Electromagnetic Wave Propagation
with st.expander("**Week 2-3: Electromagnetic Wave Propagation: Polarization, Dispersion**"):
    st.markdown("""
    ### Topics Covered
    - Wave Equation: Derivation of the electromagnetic wave equation from Maxwell’s equations.
    - Polarization: Linear, circular, and elliptical polarization; manipulation of polarized waves.
    - Dispersion: Understanding phase velocity, group velocity, and wave dispersion in different media.

    ### Problems
    7. Derive the wave equation for an electromagnetic field in a dielectric medium.
    8. Solve for the electric and magnetic fields of a plane wave propagating through free space.
    9. Analyze the polarization state of an electromagnetic wave passing through a birefringent medium.
    10. Derive the dispersion relation for electromagnetic waves in a plasma.
    11. Calculate the group and phase velocities for an electromagnetic wave propagating in a dispersive medium.
    12. Solve for the reflection and transmission coefficients of an electromagnetic wave incident on a dielectric boundary.
    13. Calculate the power carried by an electromagnetic wave in a vacuum using the Poynting vector.
    14. Determine the behavior of circularly polarized light when passing through an optically active medium.
    15. Analyze the propagation of an electromagnetic wave in a conductor and calculate the skin depth.
    16. Solve for the electromagnetic fields of a plane wave propagating through a conducting medium.
    """)

# Week 4-5: Waveguides and Resonant Cavities
with st.expander("**Week 4-5: Waveguides and Resonant Cavities**"):
    st.markdown("""
    ### Topics Covered
    - Waveguides: TE (transverse electric), TM (transverse magnetic), and TEM (transverse electromagnetic) modes in waveguides.
    - Resonant Cavities: Modes, eigenfrequencies, and boundary conditions in resonant cavities.
    - Energy Storage and Loss: Quality factor of cavities and waveguides.

    ### Problems
    17. Derive the TE and TM modes of a rectangular waveguide and calculate the cutoff frequency.
    18. Solve for the electromagnetic fields in a cylindrical waveguide and determine the modes that propagate.
    19. Calculate the resonant frequencies for a rectangular cavity and analyze the boundary conditions.
    20. Derive the quality factor for a resonant cavity with conducting walls.
    21. Solve for the wave impedance in a rectangular waveguide operating in the TE mode.
    22. Calculate the power transmitted through a waveguide as a function of the mode and frequency.
    23. Solve for the fields inside a resonant cavity with mixed boundary conditions.
    24. Analyze the energy stored in a resonant cavity and calculate the energy loss due to imperfect conductors.
    25. Derive the expression for the cutoff wavelength in a rectangular waveguide.
    26. Compute the electromagnetic field distribution inside a cavity operating in the TM mode.
    """)

# Week 6: Optical Phenomena
with st.expander("**Week 6: Optical Phenomena: Interference, Diffraction, and Polarization**"):
    st.markdown("""
    ### Topics Covered
    - Interference: Understanding interference of light waves; examples include Young’s double-slit experiment.
    - Diffraction: Fraunhofer and Fresnel diffraction; diffraction patterns from apertures and obstacles.
    - Polarization: The use of polarizers, birefringence, and optical rotation.

    ### Problems
    27. Derive the interference pattern for a double-slit experiment with coherent light sources.
    28. Solve for the diffraction pattern of light passing through a single slit using Fraunhofer diffraction.
    29. Calculate the intensity of light after passing through a polarizer and an analyzer at different angles.
    30. Analyze the interference pattern formed by a thin film and derive the conditions for constructive and destructive interference.
    31. Solve for the diffraction pattern produced by a circular aperture.
    32. Calculate the phase shift introduced by a birefringent material and determine the resulting polarization state.
    33. Analyze the diffraction pattern for a diffraction grating and compute the angles of diffraction for different orders.
    34. Solve for the transmission and reflection of polarized light through a series of polarizers at various angles.
    """)

# Week 7-8: Electromagnetic Radiation
with st.expander("**Week 7-8: Electromagnetic Radiation: Retarded Potentials, Lienard-Wiechert Potentials**"):
    st.markdown("""
    ### Topics Covered
    - Retarded Potentials: Deriving and understanding scalar and vector potentials in time-dependent situations.
    - Lienard-Wiechert Potentials: Potentials and fields from moving charges, including radiation.
    - Radiation from Accelerating Charges: Dipole radiation, quadrupole radiation, and radiation patterns.

    ### Problems
    35. Derive the retarded scalar and vector potentials for a moving point charge.
    36. Compute the Lienard-Wiechert potentials for a uniformly moving charge and derive the corresponding electric and magnetic fields.
    37. Solve for the radiation field produced by a harmonically oscillating dipole.
    38. Calculate the total power radiated by a point charge undergoing uniform acceleration.
    39. Derive the angular distribution of the radiation emitted by a dipole antenna.
    40. Compute the radiation intensity of a dipole antenna as a function of distance and angle.
    41. Derive the electric and magnetic fields of a moving point charge using the Lienard-Wiechert potentials.
    42. Analyze the radiation emitted by a charge moving in a circular orbit and derive the synchrotron radiation.
    43. Calculate the electric field of a moving dipole at a point far from the source.
    44. Derive the fields of a charge moving with uniform velocity in the near and far zones.
    """)

# Week 9: Radiation from Moving Charges
with st.expander("**Week 9: Radiation from Moving Charges**"):
    st.markdown("""
    ### Topics Covered
    - Power radiated by accelerating charges: Larmor’s formula.
    - Angular distribution of radiation.
    - Synchrotron radiation and bremsstrahlung.

    ### Problems
    45. Derive Larmor’s formula for the total power radiated by a non-relativistic accelerating charge.
    46. Calculate the angular distribution of radiation from an accelerating charge and compare it to the dipole radiation pattern.
    47. Compute the radiation emitted by a charged particle undergoing circular motion (synchrotron radiation).
    48. Analyze bremsstrahlung radiation emitted when a high-energy electron is decelerated by a nucleus.
    49. Solve for the energy spectrum of synchrotron radiation from relativistic particles in a magnetic field.
    50. Derive the expression for the radiation reaction force experienced by an accelerating charge.
    51. Calculate the radiation emitted by a relativistic electron spiraling in a magnetic field.
    52. Analyze the radiation fields produced by a charge oscillating in a harmonic potential.

    ### Topics Covered
    - Lorentz Transformations: Understanding four-vectors and Lorentz transformations in special relativity.
    - Electromagnetic Field Tensor: Covariant formulation of Maxwell’s equations using the field tensor.
    - Relativistic Electromagnetic Fields: Transformations of electric and magnetic fields under Lorentz transformations.

    ### Problems
    53. Derive the Lorentz transformations for the electric and magnetic fields between two inertial frames.
    54. Solve for the electromagnetic field tensor and show how Maxwell’s equations are expressed in covariant form.
    55. Calculate the electric and magnetic fields of a moving charge using the relativistic Lienard-Wiechert potentials.
    56. Derive the four-potential for a moving point charge and show how it transforms under Lorentz transformations.
    57. Solve for the transformation of the electromagnetic wave equation under Lorentz boosts.
    58. Derive the energy-momentum tensor for the electromagnetic field and show how it leads to conservation laws.
    59. Analyze the relativistic Doppler effect for an electromagnetic wave moving towards and away from a source.
    60. Calculate the relativistic field transformation for a charged particle moving at near-light speed.
    """)

# Week 12-13: Applications of Electromagnetic Theory to Modern Physics
with st.expander("**Week 12-13: Applications of Electromagnetic Theory to Modern Physics**"):
    st.markdown("""
    ### Topics Covered
    - Electromagnetic Fields in Astrophysics: Applications to star formation, pulsars, and black holes.
    - Quantum Electrodynamics (QED): Basic principles and interaction of photons and electrons.
    - Electromagnetic Theory in Plasma Physics: Behavior of charged particles in a plasma; Debye shielding and plasma oscillations.

    ### Problems
    61. Solve for the electromagnetic field around a neutron star using Maxwell’s equations and relativistic corrections.
    62. Analyze the interaction between a photon and an electron in a basic quantum electrodynamics (QED) process, such as Compton scattering.
    63. Calculate the behavior of electromagnetic waves in a plasma and derive the dispersion relation.
    64. Solve for the radiation emitted by a rapidly spinning pulsar using relativistic electromagnetic theory.
    65. Analyze the interaction between the cosmic microwave background radiation and charged particles in space.
    66. Derive the electromagnetic field generated by a rotating black hole using the Kerr metric.
    67. Solve for the energy lost to radiation in the early universe and its effects on the cosmic expansion.
    68. Compute the fields produced by a magnetar and analyze the effects of extreme magnetic fields on the surrounding plasma.
    69. Analyze the behavior of electromagnetic fields in a high-energy astrophysical plasma.
    70. Solve for the electromagnetic fields of a charged particle moving in the strong gravitational field of a black hole.
    """)

# Week 14: Advanced Topics in Plasma Physics and Magnetohydrodynamics
with st.expander("**Week 14: Advanced Topics: Plasma Physics and Magnetohydrodynamics (MHD)**"):
    st.markdown("""
    ### Topics Covered
    - Plasma Physics: Understanding Debye shielding, plasma oscillations, and wave propagation in plasmas.
    - Magnetohydrodynamics (MHD): Application of MHD equations to plasma flows and astrophysical systems.
    - Alfven Waves: Study of MHD waves and their role in plasma dynamics.

    ### Problems
    71. Derive the dispersion relation for plasma oscillations in a cold plasma and interpret its physical significance.
    72. Solve for the Debye length in a plasma and analyze its implications for electrostatic shielding.
    73. Compute the Alfven wave velocity in a magnetized plasma and analyze its behavior in an MHD system.
    74. Derive the MHD equations for a plasma and solve for the magnetic pressure and tension forces.
    75. Analyze the stability of an MHD pinch and solve for the conditions for stability.
    76. Solve for the energy density and flux of Alfven waves in a conducting plasma.
    77. Derive the equations governing the propagation of electromagnetic waves in a hot plasma.
    78. Solve for the effects of magnetic reconnection on the stability of an astrophysical plasma.
    """)

# Week 15: Review and Final Exam Preparation
with st.expander("**Week 15: Review and Final Exam Preparation**"):
    st.markdown("""
    ### Topics Covered
    - Comprehensive Review: Review of electromagnetic wave theory, radiation, and covariant formulation of Maxwell’s equations.
    - Problem-Solving Sessions: Focus on solving complex problems that integrate multiple course concepts.
    - Final Exam Preparation: Students will receive guidance on tackling exam problems, with a focus on key concepts and application-based questions.

    ### Problems
    79. Solve for the propagation of an electromagnetic wave in a dispersive medium and compare phase and group velocities.
    80. Analyze the radiation pattern and power emitted by a relativistic electron spiraling in a magnetic field.
    """)

# Textbooks Section
st.markdown("""
## Textbooks
- **Classical Electrodynamics** by John David Jackson  
- **Introduction to Electrodynamics** by David J. Griffiths
""")