1.1 Maxwell's equations

Maxwell's equations are the foundation of electromagnetism, describing how electric and magnetic fields interact. These four equations unify electricity and magnetism, explaining the behavior of electromagnetic waves.

Understanding Maxwell's equations is crucial for studying metamaterials and photonic crystals. They describe how fields propagate through materials, allowing us to manipulate light in novel ways and design structures with unique electromagnetic properties.

Maxwell's equations overview

• Maxwell's equations form the foundation of classical electromagnetism, describing the relationships between electric and magnetic fields, charges, and currents
• The four equations, named after James Clerk Maxwell, unify the concepts of electricity and magnetism into a coherent theory of electromagnetism
• Understanding Maxwell's equations is crucial for studying the behavior of electromagnetic waves in metamaterials and photonic crystals

Gauss's law for electric fields

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• Relates the electric flux through a closed surface to the total electric charge enclosed within that surface
• Mathematically expressed as $\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$, where $\mathbf{E}$ is the electric field, $d\mathbf{A}$ is the infinitesimal area element, $Q_{\text{enc}}$ is the enclosed charge, and $\varepsilon_0$ is the permittivity of free space
• Implies that electric field lines originate from positive charges and terminate on negative charges
• Demonstrates that the total electric flux through a closed surface is proportional to the net charge enclosed within that surface

Gauss's law for magnetic fields

• States that the magnetic flux through any closed surface is always zero
• Expressed as $\oint \mathbf{B} \cdot d\mathbf{A} = 0$, where $\mathbf{B}$ is the magnetic field and $d\mathbf{A}$ is the infinitesimal area element
• Implies that magnetic fields do not originate or terminate at any point, and magnetic field lines always form closed loops
• Highlights the fundamental difference between electric and magnetic fields, as there are no magnetic monopoles

Faraday's law of induction

• Describes how a time-varying magnetic field induces an electric field
• Mathematically expressed as $\oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}$, where $\mathbf{E}$ is the induced electric field, $d\mathbf{l}$ is the infinitesimal line element, and $\Phi_B$ is the magnetic flux
• Explains the operation of transformers, generators, and other devices that rely on electromagnetic induction
• Forms the basis for understanding the interaction between electric and magnetic fields in time-varying scenarios

Ampère's circuital law with Maxwell's addition

• Relates the magnetic field around a closed loop to the electric current and the time-varying electric flux through the loop
• Expressed as $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}$, where $\mathbf{B}$ is the magnetic field, $d\mathbf{l}$ is the infinitesimal line element, $I_{\text{enc}}$ is the enclosed current, $\mu_0$ is the permeability of free space, $\varepsilon_0$ is the permittivity of free space, and $\Phi_E$ is the electric flux
• Maxwell's addition of the displacement current term $\mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}$ ensures the consistency of Ampère's law with the continuity equation for electric charge
• Demonstrates the symmetry between electric and magnetic fields, as a time-varying electric field can generate a magnetic field

Differential form of Maxwell's equations

• The differential form of Maxwell's equations expresses the relationships between electric and magnetic fields, charges, and currents in terms of partial derivatives
• Useful for analyzing electromagnetic phenomena at a point in space and time
• Provides a more compact and mathematically convenient representation of Maxwell's equations compared to the integral form

Gauss's law in differential form

• Expressed as $\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}$, where $\mathbf{E}$ is the electric field, $\rho$ is the electric charge density, and $\varepsilon_0$ is the permittivity of free space
• Relates the divergence of the electric field to the electric charge density at a given point
• Implies that electric field lines originate from positive charge densities and terminate on negative charge densities

Faraday's law in differential form

• Expressed as $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$, where $\mathbf{E}$ is the electric field and $\mathbf{B}$ is the magnetic field
• Relates the curl of the electric field to the time rate of change of the magnetic field
• Demonstrates that a time-varying magnetic field induces an electric field with a curl proportional to the rate of change of the magnetic field

Ampère's law in differential form

• Expressed as $\nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, where $\mathbf{B}$ is the magnetic field, $\mathbf{J}$ is the electric current density, $\mu_0$ is the permeability of free space, and $\varepsilon_0$ is the permittivity of free space
• Relates the curl of the magnetic field to the electric current density and the time rate of change of the electric field
• Includes the displacement current term $\mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}$, which accounts for the generation of magnetic fields by time-varying electric fields

Integral form of Maxwell's equations

• The integral form of Maxwell's equations expresses the relationships between electric and magnetic fields, charges, and currents in terms of integrals over surfaces and paths
• Useful for analyzing electromagnetic phenomena in extended regions of space and for deriving conservation laws
• Provides a more intuitive and physically interpretable representation of Maxwell's equations compared to the differential form

Gauss's law in integral form

• Expressed as $\oint \mathbf{E} \cdot d\mathbf{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}$, where $\mathbf{E}$ is the electric field, $d\mathbf{A}$ is the infinitesimal area element, $Q_{\text{enc}}$ is the enclosed charge, and $\varepsilon_0$ is the permittivity of free space
• Relates the electric flux through a closed surface to the total electric charge enclosed within that surface
• Useful for calculating the electric field produced by symmetric charge distributions (spheres, cylinders, and planes)

Faraday's law in integral form

• Expressed as $\oint \mathbf{E} \cdot d\mathbf{l} = -\frac{d\Phi_B}{dt}$, where $\mathbf{E}$ is the induced electric field, $d\mathbf{l}$ is the infinitesimal line element, and $\Phi_B$ is the magnetic flux
• Relates the electromotive force (EMF) induced in a closed loop to the time rate of change of the magnetic flux through the loop
• Useful for analyzing electromagnetic induction in transformers, generators, and other devices

Ampère's law in integral form

• Expressed as $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{\text{enc}} + \mu_0 \varepsilon_0 \frac{d\Phi_E}{dt}$, where $\mathbf{B}$ is the magnetic field, $d\mathbf{l}$ is the infinitesimal line element, $I_{\text{enc}}$ is the enclosed current, $\mu_0$ is the permeability of free space, $\varepsilon_0$ is the permittivity of free space, and $\Phi_E$ is the electric flux
• Relates the magnetic field circulation around a closed loop to the electric current and the time rate of change of the electric flux through the loop
• Useful for calculating the magnetic field produced by symmetric current distributions (infinite wires, solenoids, and toroidal coils)

Constitutive relations

• Constitutive relations describe the response of a material to applied electric and magnetic fields
• They relate the electric displacement field $\mathbf{D}$ to the electric field $\mathbf{E}$, the magnetic field $\mathbf{H}$ to the magnetic flux density $\mathbf{B}$, and the current density $\mathbf{J}$ to the electric field $\mathbf{E}$
• The constitutive relations depend on the material properties, such as permittivity, permeability, and conductivity

Electric permittivity

• Permittivity $\varepsilon$ is a measure of a material's ability to store electric energy in response to an applied electric field
• The constitutive relation for the electric displacement field is $\mathbf{D} = \varepsilon \mathbf{E}$, where $\varepsilon = \varepsilon_0 \varepsilon_r$, with $\varepsilon_0$ being the permittivity of free space and $\varepsilon_r$ the relative permittivity of the material
• Materials with high permittivity (dielectrics) can store more electric energy and reduce the effective electric field within the material

Magnetic permeability

• Permeability $\mu$ is a measure of a material's ability to support the formation of magnetic fields in response to an applied magnetic field
• The constitutive relation for the magnetic field is $\mathbf{B} = \mu \mathbf{H}$, where $\mu = \mu_0 \mu_r$, with $\mu_0$ being the permeability of free space and $\mu_r$ the relative permeability of the material
• Materials with high permeability (ferromagnets) can enhance the magnetic field within the material and are used in transformers, inductors, and other magnetic devices

Conductivity

• Conductivity $\sigma$ is a measure of a material's ability to conduct electric current in response to an applied electric field
• The constitutive relation for the current density is $\mathbf{J} = \sigma \mathbf{E}$, known as Ohm's law
• Materials with high conductivity (metals) allow electric charges to flow easily, while materials with low conductivity (insulators) resist the flow of electric charges

Boundary conditions

• Boundary conditions describe the behavior of electric and magnetic fields at the interface between two different media
• They ensure the continuity of the tangential components of the electric and magnetic fields and the normal components of the electric displacement field and magnetic flux density across the boundary
• Boundary conditions are essential for solving electromagnetic problems involving multiple materials or regions

Tangential components of E and H

• The tangential components of the electric field $\mathbf{E}$ and the magnetic field $\mathbf{H}$ must be continuous across the boundary between two media
• Mathematically, $\mathbf{E}_{1t} = \mathbf{E}_{2t}$ and $\mathbf{H}_{1t} = \mathbf{H}_{2t}$, where the subscripts 1 and 2 denote the two media, and t denotes the tangential component
• Discontinuities in the tangential components of $\mathbf{E}$ and $\mathbf{H}$ would imply the existence of infinite currents or voltages at the boundary, which is physically impossible

Normal components of D and B

• The normal components of the electric displacement field $\mathbf{D}$ and the magnetic flux density $\mathbf{B}$ must be continuous across the boundary between two media
• Mathematically, $\mathbf{D}_{1n} = \mathbf{D}_{2n}$ and $\mathbf{B}_{1n} = \mathbf{B}_{2n}$, where the subscripts 1 and 2 denote the two media, and n denotes the normal component
• Discontinuities in the normal components of $\mathbf{D}$ and $\mathbf{B}$ would imply the existence of free electric charges or magnetic monopoles at the boundary, which is physically impossible for magnetic monopoles and requires special treatment for electric charges

Continuity of fields across boundaries

• The boundary conditions ensure the continuity of the electromagnetic fields across the interface between two media
• The continuity of the tangential components of $\mathbf{E}$ and $\mathbf{H}$ guarantees that the fields are well-defined and single-valued at the boundary
• The continuity of the normal components of $\mathbf{D}$ and $\mathbf{B}$ ensures that the fields satisfy the conservation laws for electric charge and magnetic flux

Wave equations derived from Maxwell's equations

• The wave equations for electromagnetic fields can be derived from Maxwell's equations by combining the differential forms of Faraday's law and Ampère's law
• The wave equations describe the propagation of electromagnetic waves in space and time, revealing the wave nature of light and other electromagnetic radiation
• The solutions to the wave equations provide insights into the properties of electromagnetic waves, such as their speed, polarization, and energy content

Electromagnetic wave propagation

• Electromagnetic waves are transverse waves consisting of oscillating electric and magnetic fields that are perpendicular to each other and to the direction of propagation
• The speed of electromagnetic waves in vacuum is given by $c = \frac{1}{\sqrt{\mu_0 \varepsilon_0}}$, which is approximately $3 \times 10^8$ m/s
• In a medium with permittivity $\varepsilon$ and permeability $\mu$, the speed of electromagnetic waves is $v = \frac{1}{\sqrt{\mu \varepsilon}}$

Wave equation for electric field

• The wave equation for the electric field $\mathbf{E}$ is given by $\nabla^2 \mathbf{E} - \mu \varepsilon \frac{\partial^2 \mathbf{E}}{\partial t^2} = 0$, where $\nabla^2$ is the Laplacian operator
• This equation describes the spatial and temporal evolution of the electric field in the absence of charges and currents
• The solutions to the electric field wave equation are plane waves, spherical waves, and other types of electromagnetic waves

Wave equation for magnetic field

• The wave equation for the magnetic field $\mathbf{B}$ is given by $\nabla^2 \mathbf{B} - \mu \varepsilon \frac{\partial^2 \mathbf{B}}{\partial t^2} = 0$
• This equation describes the spatial and temporal evolution of the magnetic field in the absence of charges and currents
• The solutions to the magnetic field wave equation are similar to those of the electric field wave equation, with the magnetic field being perpendicular to the electric field

Poynting vector and energy flow

• The Poynting vector is a quantity that describes the direction and magnitude of energy flow in an electromagnetic field
• It is defined as the cross product of the electric and magnetic fields, $\mathbf{S} = \mathbf{E} \times \mathbf{H}$, and has units of power per unit area (W/m^2)
• The Poynting vector is essential for understanding the energy transport and dissipation in electromagnetic systems, including metamaterials and photonic crystals

Poynting vector definition

• The Poynting vector $\mathbf{S}$ is given by $\mathbf{S} = \mathbf{E} \times \mathbf{H}$, where $\mathbf{E}$ is the electric field and $\mathbf{H}$ is the magnetic field
• The direction of the Poynting vector indicates the direction of energy flow, while its magnitude represents the power density (energy flux) at a given point
• The Poynting vector is a conserved quantity, satisfying the Poynting theorem, which relates the energy flux to the work done by the electromagnetic fields and the change in the electromagnetic energy density

Energy density of electromagnetic fields

• The energy density of an electromagnetic field is the sum of the electric and magnetic energy densities
• The electric energy density is given by $u_e = \frac{1}{2} \mathbf{D} \cdot \mathbf{E} = \frac{1}{2} \varepsilon |\mathbf{E}|^2$, while the magnetic energy density is given by $u_m = \frac{1}{2} \mathbf{B} \cdot \mathbf{H} = \frac{1}{2} \mu |\mathbf{H}|^2$
• The total electromagnetic energy density is $u = u_