pages/20_PHYf_611_Electromagnetic_Theory_I.py

import streamlit as st

# Set the page title
st.title("PHY 611: Electromagnetic Theory I")

# Course Details
st.markdown("""
## Course Details
- **Course Title**: Electromagnetic Theory I  
- **Credits**: 3  
- **Prerequisites**: PHY 416G, MA 214  
- **Instructor**: [Instructor Name]  
- **Office Hours**: [Office Hours]  

## Course Description
This course covers the fundamentals of electromagnetism, including electrostatics, boundary value problems, potential theory, energy in electromagnetic fields, and Maxwell's equations. Emphasis is placed on both mathematical rigor and physical applications in classical and modern physics.

## Course Objectives
Upon completing the course, students will be able to:
- Apply Maxwell’s equations to a wide range of physical systems.
- Solve boundary value problems in electrostatics and magnetostatics.
- Understand and analyze the behavior of electromagnetic waves in different media.

---
""")

# Weekly Outline with Problems

# Week 1-2: Introduction to Electrostatics
with st.expander("**Week 1-2: Introduction to Electrostatics (Gauss’s Law, Electric Field, and Potential)**"):
    st.markdown("""
    ### Topics Covered
    - Gauss’s Law: Introduction to the divergence of the electric field, integral and differential forms of Gauss’s law.
    - Electric Field: Electric fields due to point charges, continuous charge distributions, and line charges.
    - Electric Potential: Concept of potential energy in electrostatics, relationship between electric field and potential.
    - Applications of Gauss’s Law: Symmetry in electrostatic problems (spherical, cylindrical, and planar symmetry).

    ### Problems
    1. Use Gauss’s law to calculate the electric field of a spherical charge distribution.
    2. Calculate the electric field of an infinitely long charged line using Gauss’s law.
    3. Derive the electric field for a uniformly charged infinite plane.
    4. Find the potential due to a point charge at an arbitrary distance.
    5. Calculate the electric potential for a uniformly charged spherical shell.
    6. Determine the electric field inside and outside a uniformly charged solid sphere.
    7. Apply Gauss’s law to find the electric field of a charged conducting sphere.
    8. Solve for the electric field and potential for a continuous charge distribution along a line.
    9. Calculate the potential energy of a system of two point charges.
    10. Compute the energy stored in an electric field created by a system of charges.
    """)

# Week 3-4: Solutions to Laplace’s Equation
with st.expander("**Week 3-4: Solutions to Laplace’s Equation (Separation of Variables, Multipole Expansion)**"):
    st.markdown("""
    ### Topics Covered
    - Laplace’s Equation: Introduction to Laplace’s and Poisson’s equations in Cartesian, spherical, and cylindrical coordinates.
    - Separation of Variables: Solution techniques for Laplace’s equation in different coordinate systems.
    - Multipole Expansion: Concept of the monopole, dipole, and higher-order terms in electrostatics.
    - Boundary Conditions: Continuity of potential and electric field across boundaries.

    ### Problems
    11. Solve Laplace’s equation for a potential in Cartesian coordinates with specific boundary conditions.
    12. Use separation of variables to solve Laplace’s equation in spherical coordinates.
    13. Solve Laplace’s equation in cylindrical coordinates for a system with cylindrical symmetry.
    14. Derive the potential outside a charge distribution using multipole expansion.
    15. Calculate the quadrupole moment for a charge distribution and find its contribution to the potential.
    16. Apply boundary conditions to solve Laplace’s equation in a rectangular box.
    17. Use spherical harmonics to solve the potential of a charge located off-center inside a sphere.
    18. Derive the dipole potential at a point far from a given charge distribution.
    19. Solve for the potential inside a conducting shell with a charge placed inside it using Laplace’s equation.
    20. Compute the monopole, dipole, and quadrupole contributions to the potential of a distant charge distribution.
    """)

# Week 5-6: Conductors and Dielectrics
with st.expander("**Week 5-6: Conductors and Dielectrics in Electrostatic Fields**"):
    st.markdown("""
    ### Topics Covered
    - Conductors: Behavior of conductors in electrostatic equilibrium, boundary conditions at conductor surfaces.
    - Electrostatic Shielding: Concept and applications of shielding.
    - Capacitance: Calculating capacitance for various geometries.
    - Dielectrics: Polarization, bound charges, and the electric displacement field \( \mathbf{D} \).
    - Gauss’s Law in Dielectrics: Modified form of Gauss's law in dielectric media.

    ### Problems
    21. Calculate the capacitance of a parallel plate capacitor with and without a dielectric.
    22. Solve for the electric field inside and outside a conducting sphere placed in an external uniform electric field.
    23. Determine the potential and electric field in a spherical capacitor with a dielectric.
    24. Use boundary conditions to solve for the electric displacement field \( \mathbf{D} \) in a dielectric material.
    25. Solve for the charge distribution on a conductor placed near another charged object.
    26. Calculate the polarization vector for a dielectric material in a uniform electric field.
    27. Analyze the behavior of conductors in electrostatic equilibrium, including the surface charge distribution.
    28. Determine the energy stored in a system of capacitors, both with and without dielectrics.
    """)

# Week 7-8: Magnetostatics
with st.expander("**Week 7-8: Magnetostatics (Ampère’s Law, Vector Potential)**"):
    st.markdown("""
    ### Topics Covered
    - Ampère’s Law: Integral and differential forms of Ampère’s law in the static case.
    - Magnetic Fields from Currents: The Biot-Savart law, magnetic field due to various current distributions.
    - Vector Potential: Introduction to the vector potential \( \mathbf{A} \), and its relation to magnetic fields.
    - Boundary Conditions: Behavior of magnetic fields at boundaries between media.

    ### Problems
    29. Use Ampère’s law to calculate the magnetic field inside a long solenoid.
    30. Apply Ampère’s law to find the magnetic field due to a current-carrying wire.
    31. Derive the magnetic field of a toroidal coil using Ampère’s law.
    32. Solve for the magnetic field of a straight wire using the Biot-Savart law.
    33. Derive the vector potential \( \mathbf{A} \) for a circular current loop.
    34. Calculate the magnetic dipole moment for a current loop and its field far away.
    35. Determine the boundary conditions for the magnetic field at the surface of a conductor.
    36. Use the concept of vector potential to find the magnetic field in a cylindrical current distribution.
    37. Solve for the magnetic field inside and outside a coaxial cable using Ampère’s law.
    38. Compute the force between two parallel current-carrying wires.
    """)

# Week 9: Maxwell’s Equations
with st.expander("**Week 9: Maxwell’s Equations and Energy in Electromagnetic Fields**"):
    st.markdown("""
    ### Topics Covered
    - Maxwell’s Equations: Time-dependent form of Maxwell’s equations, introduction to displacement current.
    - Conservation Laws: Charge conservation and the continuity equation.
    - Energy in Electromagnetic Fields: Poynting vector and the energy flux in electromagnetic fields.
    - Electromagnetic Momentum: Momentum in fields and forces on charges.

    ### Problems
    39. Derive Maxwell’s equations in the presence of free and bound charges.
    40. Apply Maxwell’s equations to a static electric field in a vacuum.
    41. Derive the Poynting vector from Maxwell’s equations and explain its physical meaning.
    42. Solve for the energy density stored in an electromagnetic field.
    43. Use Maxwell’s equations to derive the wave equation for electric and magnetic fields in free space.
    44. Compute the energy flux of an electromagnetic field using the Poynting theorem.
    45. Apply Maxwell’s equations to a time-varying electric field and solve for the induced magnetic field.
    46. Calculate the force exerted by an electromagnetic field on a moving charge distribution.
    """)

# Week 10-11: Electromagnetic Waves
with st.expander("**Week 10-11: Electromagnetic Waves (Wave Equations, Energy, and Momentum)**"):
    st.markdown("""
    ### Topics Covered
    - Wave Equations: Derivation of the wave equation from Maxwell’s equations in vacuum and in media.
    - Plane Waves: Solutions to the wave equation for plane electromagnetic waves.
    - Polarization: Linear, circular, and elliptical polarization of light waves.
    - Energy and Momentum in Waves: Energy density and momentum density of electromagnetic waves.

    ### Problems
    47. Derive the wave equation for an electromagnetic field in vacuum from Maxwell’s equations.
    48. Solve for the electric and magnetic fields of a plane wave propagating in free space.
    49. Determine the polarization state of an electromagnetic wave and describe circular polarization.
    50. Calculate the energy density and momentum density of a plane electromagnetic wave.
    51. Analyze the propagation of an electromagnetic wave in a dielectric medium.
    52. Derive the expression for the intensity of an electromagnetic wave in a vacuum.
    53. Solve for the wave equation in a conducting medium and calculate the skin depth.
    54. Compute the energy carried by an electromagnetic wave in a vacuum and its rate of energy transfer.
    55. Calculate the Doppler shift for electromagnetic waves in different frames of reference.
    56. Analyze the reflection and transmission of electromagnetic waves at the interface between two dielectric media.
    """)

# Week 12: Reflection and Transmission of Electromagnetic Waves
with st.expander("**Week 12: Reflection and Transmission of Electromagnetic Waves**"):
    st.markdown("""
    ### Topics Covered
    - Fresnel Equations: Derivation and application of Fresnel's equations for reflection and transmission at boundaries.
    - Brewster’s Angle: Condition for no reflection and its physical significance.
    - Total Internal Reflection: Critical angle and total internal reflection.
    - Skin Depth: Electromagnetic wave penetration in conductors and skin depth phenomena.

    ### Problems
    57. Derive Fresnel’s equations for reflection and transmission at a dielectric interface.
    58. Solve for the reflection and transmission coefficients for a wave incident at Brewster’s angle.
    59. Compute the total internal reflection angle for an electromagnetic wave propagating in a medium.
    60. Calculate the intensity of the transmitted wave through a boundary between two media with different refractive indices.
    61. Analyze the phase change upon reflection of a wave at a dielectric interface.
    62. Solve for the skin depth of an electromagnetic wave in a conductor and calculate the energy lost.
    63. Use Snell's law to derive the critical angle for total internal reflection.
    64. Analyze the electromagnetic wave’s transmission through a multi-layer dielectric system.
    """)

# Week 13: Special Relativity and Electromagnetism
with st.expander("**Week 13: Special Relativity and Electromagnetism**"):
    st.markdown("""
    ### Topics Covered
    - Lorentz Transformations: Introduction to special relativity and Lorentz transformations.
    - Four-Vectors: Space-time and four-vector notation.
    - Electromagnetic Field Tensor: Covariant formulation of Maxwell’s equations.
    - Relativistic Electromagnetic Fields: Fields of moving charges, relativistic transformations of fields.

    ### Problems
    65. Derive the Lorentz transformation for electric and magnetic fields under a change in reference frames.
    66. Compute the fields of a moving point charge using the Lorentz transformations.
    67. Use four-vectors to express the current density and charge density in a relativistic framework.
    68. Solve for the electric and magnetic fields of a uniformly moving charge.
    69. Derive the electromagnetic field tensor and use it to rewrite Maxwell’s equations in covariant form.
    70. Calculate the relativistic field transformation of an electromagnetic wave under a boost.
    71. Analyze the relativistic Doppler effect for electromagnetic radiation.
    72. Use the invariance of the electric and magnetic fields to solve a relativistic boundary condition problem.
    """)

# Week 14: Radiation (Dipole Radiation and Scattering)
with st.expander("**Week 14: Radiation (Dipole Radiation and Scattering)**"):
    st.markdown("""
    ### Topics Covered
    - Radiation from Accelerated Charges: Larmor’s formula and energy radiated by accelerating charges.
    - Dipole Radiation: Derivation and physical interpretation of dipole radiation.
    - Scattering: Rayleigh scattering and its role in optics.
    - Antenna Theory: Radiation from dipole antennas and applications in telecommunications.

    ### Problems
    73. Derive the power radiated by an oscillating electric dipole using Larmor’s formula.
    74. Solve for the angular distribution of radiation emitted by a dipole antenna.
    75. Analyze the polarization of dipole radiation and compute the radiated power in a specific direction.
    76. Calculate the scattering cross-section for Rayleigh scattering.
    77. Derive the radiation pattern for a magnetic dipole.
    78. Solve for the total radiated power from a system of two oscillating charges.
    79. Apply Larmor’s formula to calculate the radiation emitted by a particle in circular motion.
    80. Solve for the radiation intensity of a dipole antenna as a function of distance and angle.
    """)

# Week 15: Review and Final Exam Preparation
with st.expander("**Week 15: Review and Final Exam Preparation**"):
    st.markdown("""
    ### Topics Covered
    - Review of Maxwell’s Equations: Comprehensive review of Maxwell’s equations and their applications.
    - Problem-Solving Sessions: Review of key problem-solving techniques used throughout the course.
    - Sample Problems: Practice with complex electromagnetic scenarios to prepare for the final exam.
    - Final Exam Preparation: Focused review on boundary value problems, wave phenomena, and radiation theory.
    """)

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