electromagnetism

# Electromagnetism ## What I Do I provide comprehensive electromagnetism tools including electrostatic fields, magnetic fields, Maxwell's equations, electromagnetic waves, radiation theory, and circuit analysis for physics and engineering applications. ## When to Use Me - Electric field and potential calculations - Magnetic field analysis - Electromagnetic wave propagation - Radiation and antenna theory - Circuit analysis - Plasma physics ## Core Concepts - **Electrostatics**: Coulomb's law, Gauss's law, Poisson's equation - **Magnetostatics**: Biot-Savart law, Ampere's law - **Maxwell's Equations**: Integral and differential forms - **EM Waves**: Wave equation, polarization, propagation - **Potentials**: Scalar and vector potentials, gauge invariance - **Radiation**: Dipole radiation, antenna patterns - **Boundary Conditions**: Dielectric and conductor interfaces - **Electromagnetic Materials**: Permittivity, permeability ## Code Examples ### Electrostatic Fields ```python import numpy as np k_e = 8.99e9 # Coulomb constant def electric_field_point_charge(q, r, r_vec): return k_e * q * r_vec / np.linalg.norm(r_vec)**3 def electric_potential_point_charge(q, r): return k_e * q / r def superposition_e_field(charges, positions, observation_point): E = np.zeros(3) for q, r in zip(charges, positions): r_vec = observation_point - r r_mag = np.linalg.norm(r_vec) E += k_e * q * r_vec / r_mag**3 return E charges = [1e-6, -1e-6] positions = [np.array([0, 0, 0]), np.array([0.1, 0, 0])] E = superposition_e_field(charges, positions, np.array([0.05, 0.05, 0])) print(f"Electric field: {E}") ``` ### Gauss's Law ```python def electric_flux_through_surface(E, dA): return np.sum(E * dA) def enclosed_charge_from_flux(flux, epsilon=8.85e-12): return epsilon * flux # Dipole moment def dipole_moment(q, d): return q * d def field_on_axis_dipole(p, r, epsilon=8.85e-12): k = 1 / (4 * np.pi * epsilon) return 2 * k * p / r**3 p = 1e-9 * np.array([0.01, 0, 0]) r = 0.1 E_axis = field_on_axis_dipole(p, r) print(f"Field on dipole axis: {E_axis}") ``` ### Magnetic Fields ```python mu_0 = 4e-7 * np.pi # Permeability of free space def biot_savart_field(I, dl, r_obs, r_source): r_vec = r_obs - r_source r_mag = np.linalg.norm(r_vec) return mu_0 / (4 * np.pi) * I * np.cross(dl, r_vec) / r_mag**3 def magnetic_dipole_field(m, r): r_mag = np.linalg.norm(r) return mu_0 / (4 * np.pi) * (3 * np.dot(m, r) * r / r_mag**5 - m / r_mag**3) m = np.array([0, 0, 1e-3]) r = np.array([0.1, 0.1, 0]) B = magnetic_dipole_field(m, r) print(f"Magnetic field from dipole: {B}") ``` ### Maxwell's Equations ```python def faraday_law(dB_dt, area): return -dB_dt * area def ampere_maxwell_law(I, dE_dt, epsilon=8.85e-12, mu=4e-7*np.pi): return I + epsilon * mu * dE_dt * area def wave_equation_coefficients(epsilon, mu, sigma=0): c = 1 / np.sqrt(epsilon * mu) alpha = sigma / (2 * epsilon) return c, alpha epsilon = 8.85e-12 c, alpha = wave_equation_coefficients(epsilon, 4e-7*np.pi) print(f"Speed of light in medium: {c:.2e} m/s") print(f"Attenuation constant: {alpha:.2e}") ``` ### Electromagnetic Waves ```python def wave_impedance(epsilon, mu): return np.sqrt(mu / epsilon) def skin_depth(sigma, omega, mu, epsilon): return np.sqrt(2 / (omega * mu * sigma)) def reflected_power(n1, n2): return ((n2 - n1) / (n2 + n1))**2 epsilon_r = 2.1 mu_r = 1 sigma = 1e-2 f = 1e9 eta = wave_impedance(epsilon_r * 8.85e-12, mu_r * 4e-7*np.pi) delta = skin_depth(sigma, 2*np.pi*f, mu_r * 4e-7*np.pi, epsilon_r * 8.85e-12) print(f"Wave impedance: {eta:.2f} Ω") print(f"Skin depth: {delta:.2e} m") ``` ## Best Practices 1. **Boundary Conditions**: Apply appropriate BCs at interfaces 2. **Singularities**: Handle point charges carefully 3. **Units**: Use SI units consistently 4. **Gauge Choice**: Choose appropriate gauge for potentials 5. **Materials**: Account for frequency-dependent properties ## Common Patterns ```python # Poynting vector def poynting_vector(E, H): return np.cross(E, np.conj(H)) # Radiation resistance def radiation_resistance(I, l, f, c=3e8): return 80 * np.pi**2 * (I * l / c)**2 * (f/c)**2 # Retarded potentials def retarded_time(t_obs, r, c): return t_obs - np.linalg.norm(r) / c ``` ## Core Competencies 1. Electrostatic and magnetostatic fields 2. Maxwell's equations and wave propagation 3. Boundary value problems 4. Electromagnetic radiation 5. Circuit and transmission line theory