13.2 Maxwell's equations in integral and differential forms
4 min read•august 7, 2024
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
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Gauss’s Law for Electric Fields — Electromagnetic Geophysics View original
for electricity relates the electric flux through a closed surface to the total electric charge enclosed within that surface
Mathematically expressed as ∮E⋅dA=ϵ0Qenc, where E is the , dA is the area element, Qenc is the enclosed charge, and ϵ0 is the of free space
Gauss's law for magnetism states that the magnetic flux through any closed surface is always zero
Mathematically expressed as ∮B⋅dA=0, where B is the and dA 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 ∇⋅E=ϵ0ρ, where ρ is the volume charge density
Relates the 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 ∇⋅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 describes how a changing magnetic flux induces an electromotive force (EMF) in a loop of wire
Mathematically expressed as ∮E⋅dl=−dtdΦB, where E is the electric field, dl is the line element, and ΦB 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 ∮B⋅dl=μ0Ienc+μ0ϵ0dtdΦE, where B is the magnetic field, dl is the line element, Ienc is the enclosed current, μ0 is the of free space, ϵ0 is the permittivity of free space, and ΦE is the electric flux
The term μ0ϵ0dtdΦE 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 ∇×E=−∂t∂B, which relates the 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 ∇×B=μ0J+μ0ϵ0∂t∂E, where 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 ∇ is a vector differential operator used in vector calculus
In Cartesian coordinates, ∇=i^∂x∂+j^∂y∂+k^∂z∂, where i^, j^, and k^ are unit vectors in the x, y, and z directions, respectively
Used to define the , divergence, and curl of a vector field
Curl and Divergence
The curl of a vector field F is defined as ∇×F and measures the infinitesimal rotation of the field
In Cartesian coordinates, ∇×F=(∂y∂Fz−∂z∂Fy)i^+(∂z∂Fx−∂x∂Fz)j^+(∂x∂Fy−∂y∂Fx)k^
The divergence of a vector field F is defined as ∇⋅F and measures the infinitesimal flux of the field per unit volume
In Cartesian coordinates, ∇⋅F=∂x∂Fx+∂y∂Fy+∂z∂Fz
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