Maxwell's equations are the cornerstone of electromagnetism. They describe how electric and magnetic fields interact and evolve. These equations come in two forms: integral and differential, each offering unique insights into electromagnetic phenomena.

The integral form gives a big-picture view, showing how fields behave over larger areas or volumes. The differential form, on the other hand, zooms in on specific points, revealing local field behavior. Together, they paint a complete picture of electromagnetic theory.

Gauss's Laws

Gauss's Law for Electricity and Magnetism

• Gauss's law for electricity relates the electric flux through a closed surface to the total electric charge enclosed within that surface
• Mathematically expressed as $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$, where $\vec{E}$ is the electric field, $d\vec{A}$ is the area element, $Q_{enc}$ is the enclosed charge, and $\epsilon_0$ is the permittivity of free space
• Gauss's law for magnetism states that the magnetic flux through any closed surface is always zero
• Mathematically expressed as $\oint \vec{B} \cdot d\vec{A} = 0$, where $\vec{B}$ is the magnetic field and $d\vec{A}$ is the area element
• Implies that magnetic monopoles do not exist and magnetic field lines always form closed loops

Integral and Differential Forms of Gauss's Laws

• Integral form of Gauss's law for electricity relates the total electric flux through a closed surface to the total electric charge enclosed within that surface
• Differential form of Gauss's law for electricity is $\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$, where $\rho$ is the volume charge density
• Relates the divergence of the electric field at a point to the charge density at that point
• Integral form of Gauss's law for magnetism states that the magnetic flux through any closed surface is always zero
• Differential form of Gauss's law for magnetism is $\nabla \cdot \vec{B} = 0$
• Implies that the divergence of the magnetic field is always zero at any point in space

Faraday's and Ampère-Maxwell Laws

Faraday's Law of Induction

• Faraday's law of induction describes how a changing magnetic flux induces an electromotive force (EMF) in a loop of wire
• Mathematically expressed as $\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$, where $\vec{E}$ is the electric field, $d\vec{l}$ is the line element, and $\Phi_B$ is the magnetic flux
• Negative sign indicates that the induced EMF opposes the change in magnetic flux (Lenz's law)
• Example: A moving magnet near a coil of wire induces an electric current in the coil

Ampère-Maxwell Law

• Ampère-Maxwell law relates the magnetic field circulating around a closed loop to the electric current and the rate of change of electric flux through the loop
• Mathematically expressed as $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc} + \mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$, where $\vec{B}$ is the magnetic field, $d\vec{l}$ is the line element, $I_{enc}$ is the enclosed current, $\mu_0$ is the permeability of free space, $\epsilon_0$ is the permittivity of free space, and $\Phi_E$ is the electric flux
• The term $\mu_0 \epsilon_0 \frac{d\Phi_E}{dt}$ is Maxwell's displacement current, which accounts for the fact that a changing electric field can generate a magnetic field
• Example: A charging capacitor produces a magnetic field in the surrounding space

Integral and Differential Forms of Faraday's and Ampère-Maxwell Laws

• Integral form of Faraday's law relates the EMF induced in a closed loop to the rate of change of magnetic flux through the loop
• Differential form of Faraday's law is $\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t}$, which relates the curl of the electric field to the time rate of change of the magnetic field
• Integral form of Ampère-Maxwell law relates the magnetic field circulating around a closed loop to the electric current and the rate of change of electric flux through the loop
• Differential form of Ampère-Maxwell law is $\nabla \times \vec{B} = \mu_0 \vec{J} + \mu_0 \epsilon_0 \frac{\partial \vec{E}}{\partial t}$, where $\vec{J}$ is the current density
• Relates the curl of the magnetic field to the current density and the time rate of change of the electric field

Vector Calculus Operators

Nabla Operator

• The nabla operator $\nabla$ is a vector differential operator used in vector calculus
• In Cartesian coordinates, $\nabla = \hat{i} \frac{\partial}{\partial x} + \hat{j} \frac{\partial}{\partial y} + \hat{k} \frac{\partial}{\partial z}$, where $\hat{i}$, $\hat{j}$, and $\hat{k}$ are unit vectors in the x, y, and z directions, respectively
• Used to define the gradient, divergence, and curl of a vector field

Curl and Divergence

• The curl of a vector field $\vec{F}$ is defined as $\nabla \times \vec{F}$ and measures the infinitesimal rotation of the field
• In Cartesian coordinates, $\nabla \times \vec{F} = \left(\frac{\partial F_z}{\partial y} - \frac{\partial F_y}{\partial z}\right) \hat{i} + \left(\frac{\partial F_x}{\partial z} - \frac{\partial F_z}{\partial x}\right) \hat{j} + \left(\frac{\partial F_y}{\partial x} - \frac{\partial F_x}{\partial y}\right) \hat{k}$
• The divergence of a vector field $\vec{F}$ is defined as $\nabla \cdot \vec{F}$ and measures the infinitesimal flux of the field per unit volume
• In Cartesian coordinates, $\nabla \cdot \vec{F} = \frac{\partial F_x}{\partial x} + \frac{\partial F_y}{\partial y} + \frac{\partial F_z}{\partial z}$
• The curl and divergence appear in the differential forms of Maxwell's equations, relating the electric and magnetic fields to their sources and each other