Electricity & Magnetism

Posted on Edited on In

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magnetic field (magnetic flux density, \(B\)), is the tesla (symbol: \(T\)), defined as one weber per square meter (\(Wb/m^2\))

Magnetic flux \(\Phi_B\) measures the total magnetic field (\(B\)) passing through a given surface area (\(A\)), representing the number of field lines penetrating that area. Measured in Webers (\(Wb\)),

Vector Calculus

3Blue1Brown, Divergence and curl: The language of Maxwell's equations, fluid flow, and more [https://youtu.be/rB83DpBJQsE]

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Gradient

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Divergence

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# https://share.google/aimode/l3lNa2MRAOG8hkpOc

import numpy as np
import matplotlib.pyplot as plt

# 1. Define the grid
x = np.linspace(-6, 6, 40)
y = np.linspace(-3, 3, 20)
X, Y = np.meshgrid(x, y)

# 2. Define the vector field components F = [U, V]
U = np.sin(X)
V = np.cos(Y)

# 3. Calculate Divergence (Scalar Field)
div_F = np.cos(X) - np.sin(Y)

# 4. Create the plot
fig, ax = plt.subplots(figsize=(16, 8))

# Use 'RdBu_r' (reversed) so Positive = Red, Negative = Blue
contour = ax.contourf(X, Y, div_F, cmap='RdBu_r', levels=30, alpha=0.8)
fig.colorbar(contour, label='Divergence (Red=Source, Blue=Sink)')

# Plot Vector Field
ax.quiver(X, Y, U, V, color='black', alpha=0.9, scale=20)
ax.set_xlabel('X', fontsize=12)
ax.set_ylabel('Y', fontsize=12)
ax.set_title(r'Positive Divergence (Red) and Negative Divergence (Blue)')
plt.show()

Curl

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# https://share.google/aimode/hWb3cR4vCBoWV4Moi

import numpy as np
import matplotlib.pyplot as plt

# 1. Setup the coordinate grid
x = np.linspace(-2, 2, 7)
y = np.linspace(-2, 2, 7)
z = np.linspace(-2, 2, 7)
X, Y, Z = np.meshgrid(x, y, z)

# 2. Define the Vector Field F and Curl(F)
U, V, W = X, Y*Z, 3*X*Z
C_U, C_V, C_W = -Y, -3*Z, np.zeros_like(Z)

# 3. Create the plot
fig = plt.figure(figsize=(16, 8), constrained_layout=True)

# Subplot 1: Vector Field
ax1 = fig.add_subplot(121, projection='3d')
ax1.quiver(X, Y, Z, U, V, W, length=0.3, normalize=True, color='royalblue')
ax1.set_title('Vector Field F')
ax1.set_xlabel('X', fontsize=16)  # Adding x-label
ax1.set_ylabel('Y', fontsize=16)  # Adding y-label
ax1.set_zlabel('Z', fontsize=16)  # Adding z-label

# Subplot 2: Curl
ax2 = fig.add_subplot(122, projection='3d')
ax2.quiver(X, Y, Z, C_U, C_V, C_W, length=0.3, normalize=True, color='crimson')
ax2.set_title('Curl(F)')
ax2.set_xlabel('X', fontsize=16)  # Adding x-label
ax2.set_ylabel('Y', fontsize=16)  # Adding y-label
ax2.set_zlabel('Z', fontsize=16)  # Adding z-label

# plt.tight_layout(pad=3.0, rect=[0, 0, 1, 0.95])
plt.show()

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cross product [Google AI Mode]

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Divergence Theorem (Gauss's Theorem)

surface integral -> volume integral \[ \oint_S \vec{F} \cdot d\vec{S} = \int_V (\nabla \cdot \vec{F}) dv \]

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Divergence theorem is only applicable to closed surfaces

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Stoke's theorem

line integral -> surface inegral \[ \oint_{c} \vec{F} \cdot \vec{\mathrm{d}l} = \int_{s} (\nabla \times \vec{F}) \cdot \vec{\mathrm{d}S} \]

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Magnetostatics

Magnetostatics is the study of magnetic fields in systems where the currents are steady (not changing with time)

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Magnetic Field Intensity(\(H\)) & Magnetic Flux Density(\(B\))

TODO 📅

Magnetic Potential

Magnetic Scalar Potential

TODO 📅

Magnetic Vector Potential

TODO 📅

Electrodynamics

Faraday's Law

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Lenz's law

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Ampère's law with Maxwell's correction

displacement current

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[https://youtu.be/t-W6_fn_1bI]

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[Google AI Mode]

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Field Inside and Outside a Current-Carrying Wire

Sources of Magnetic Fields [https://web.mit.edu/8.02t/www/802TEAL3D/visualizations/coursenotes/modules/guide09.pdf]

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Displacement Current

A. Sheikholeslami, "Current Without Electrons [Circuit Intuitions]," in IEEE Solid-State Circuits Magazine, vol. 17, no. 4, pp. 8-10, Fall 2025

—, "Current Without Electric Field [Circuit Intuitions]," in IEEE Solid-State Circuits Magazine, vol. 18, no. 1, pp. 8-12, winter 2026

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energy and information are carried by electric and magnetic fields (\(E\) and \(H\)) rather than by electron drift

proximity effect & skin effect

  • Skin effect concentrates current near the surface of a single conductor, while proximity effect concentrates current in specific regions of multiple conductors due to their interaction
  • Skin effect is caused by the conductor's own magnetic field, while proximity effect is caused by the magnetic field of a nearby conductor

proximity effect is a redistribution of electric current occurring in nearby parallel electrical conductors carrying alternating current (AC), caused by magnetic effects (eddy currents)

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skin effect is the tendency of AC current flow near the surface (or "skin") of a conductor, rather than throughout its cross-section, due to the magnetic field generated by the current itself

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Cause of skin effect

A main current \(I\) flowing through a conductor induces a magnetic field \(H\). If the current increases, as in this figure, the resulting increase in \(H\) induces separate, circulating eddy currents \(I_W\) which partially cancel the current flow in the center and reinforce it near the skin

Eddy current

By Lenz's law, an eddy current creates a magnetic field that opposes the change in the magnetic field that created it, and thus eddy currents react back on the source of the magnetic field

reference

Griffiths, David J. Introduction to Electrodynamics. Fifth edition. Cambridge University Press, 2024. [pdf]

David Smith, Electromagnetic Theory for Complete Idiot, 2021

邓友金. 电磁学 2022春 [http://staff.ustc.edu.cn/~yjdeng/EM2022/EM2022.html]

谢处方、饶克谨、杨显清等.《电磁场与电磁波》（第五版），高等教育出. 版社，2019.

Scott Hughes. Spring 2005 8.022: Electricity & Magnetism [https://web.mit.edu/sahughes/www/8.022/]