The Ampere-Maxwell equation relates electric currents and magnetic flux. Itdescribes the magnetic fields that result from a transmitter wire or loop inelectromagnetic surveys. For steady currents, it is key for describing themagnetometric resistivity experiment.
Integral Equation¶
The Ampere-Maxwell equation in integral form is given below:
where:
(mathbf{b}) is the magnetic flux
(mathbf{e}) is the electric field
(I_{enc}) is the enclosed current
(mu_0) is the magnetic permeability of free space
(varepsilon_0) is the electric permittivity of free space
(hat{mathbf{n}}) is the outward pointing unit-normal
Ampere's law allows us to calculate magnetic fields from the relation between the electric currents that generate this magnetic fields. It states that for a closed path the sum over elements of the component of the magnetic field is equal to electric current multiplied by the empty's permeability. The integral form of Amperes' Circuital Law (ACL) for magnetostatics relates the magnetic field along a closed path to the total current flowing through any surface bounded by that path. 7.4: Ampere's Circuital Law (Magnetostatics) - Integral Form - Engineering LibreTexts.
The first term of the right hand side of the equation was discovered by Ampere. It shows the relationshipbetween a current (I_{enc}) and the circulation of the magnetic field, (mathbf{b}),around any closed contour line (See Fig. 36). (I_{enc}) refers to all currentsirrespective of their physical origin.
The second portion of the equation is Maxwell's contribution and shows that acirculation of magnetic field is also caused by a time rate of change ofelectric flux. This explains how current in a simple circuit involving abattery and capacitor can flow. The term is pivotal in showing thatelectromagnetic energy propagates as waves.
Fig. 37 Integration over a capacitor¶
For example, imagine integrating over a surface associated with a closed pathsuch as the one showed in Fig. 37. We can define the surface to bethe area of the circle, as in Fig. 36, or alternatively, as astretched surface, as shown in Fig. 37. In the first case,the enclosed current is the flow of charges in the wire. In the second case,however, there are no charges flowing through the surface, yet the magneticfield defined on the enclosing curve, (C), must be the same. This apparentdiscrepancy is reconciled if we take into account the displacement current,which is the time rate of change of the electric field, between the twoplates. This integration is the same as if we were integrating over a flatsurface with the current wire crossing it.
Ampere's Circuital Law Differential Form
The integral formulations are physically insightful and closely relate to theexperiments that gave rise to them. They also play a formative role ingenerating boundary conditions for waves that propagate through differentmaterials.
When dealing with the propagation of EM waves in matter the currents(I_{enc}) are usually dealt with in terms of current densities. Theintegral equation above is thus written as
where the current densities are:
(mathbf{j_f}) is the free current caused by moving charges
(mathbf{j_p} = frac{partial mathbf{p}}{partial t}) is the polarization or bound current, where (mathbf{p}) is the electric polarization resulting from bound charges in dielectrics
(mathbf{j_m} = nabla times mathbf{m}) is the magnetization current, that is, the currents needed to generate the magnetization (mathbf{m})
The total current density is the sum of these three contributions and is described by
Note
The total current involved in the Ampere-Maxwell equation consists of freecurrent and bound current, although all currents are essentially the same froma microscopic perspective. Treating free current and bound current differentlyoffers physical insights to the Ampere-Maxwell equation in different contexts.
The free current is caused by moving charges which are not tied to atoms, oftenreferred to as conduction current. In contrast, the bound current is induced bya magnetization or a polarization in bulk materials. When a magnetic material isplaced in an external magnetic field, a magnetization current will be induceddue to the motion of electrons in atoms. Likewise, when an external electricfield is applied to a dielectric material, the positive and negative bound charges withinthe dielectric can separate and induce a polarization current density internally.
Continuing to treat the free current and bound current separately and using theconstitutive equations: (mathbf{b} = mu_0(mathbf{h} + mathbf{m})) and (mathbf{d}= varepsilon_0 mathbf{e} + mathbf{p}), the integral form Ampere-Maxwell equation can be reformulated as:
Note that the bound charge due to magnetization is integrated into the magneticfield (mathbf{h}), whereas the bound charge due to electric polarization isintegrated into the displacement field (mathbf{d}).
Differential equation in the time domain¶
There are a number of ways of writing the equation in differential form. Eachprovides its own insight. We begin by considering the differential form of equation (62) in terms of the variables (mathbf{e, b, p}) and (mathbf{m}):
and similar to (65), we can use the constitutive relations (mathbf{d}= varepsilon_0 mathbf{e} + mathbf{p}) and (mathbf{b} = mu_0(mathbf{h} + mathbf{m})) to write the differential time-domain equation in terms of the variables (mathbf{h, j_f}) and (mathbf{d}):
Ampere Maxwell Law Differential Form
Differential equations in the frequency domain¶
We use the (e^{iomega t})Fourier Transform Convention to transferour equations from the time domain to the frequency domain.
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The equation becomes
If we deal with linear isotropic media then we have
and the Ampere-Maxwell equations can be written as
Units¶
Magnetic B-field | (mathbf{b}) | T | tesla |
Electric field intensity | (mathbf{e}) | (frac{text{V}}{text{m}}) | volt per meter |
Electric current | (text{I}) | A | ampere |
Electric current density | (mathbf{j}) | (frac{text{A}}{text{m}^{2}}) | ampere per square meter |
Magnetization | (mathbf{m}) | (frac{text{A}}{text{m}}) | ampere per meter |
Electric polarization | (mathbf{p}) | (frac{text{A}cdottext{s}}{text{m}}) | ampere times seconds per square meter |
Magnetic H-field | (mathbf{h}) | (frac{text{A}}{text{m}}) | ampere per meter |
Electric displacement | (mathbf{d}) | (frac{text{C}}{text{m}^{2}}) | coulomb per square meter |
Constants
Magnetic constant | (mu_0 = 4pi ×10^{−7} frac{text{N}}{text{A}^2} approx 1.2566370614..×10^{-6} frac{text{T}cdot text{m}}{text{A}}) |
Vacuum permittivity | (varepsilon_0 approx 8.854 187 817.. × 10^{−12} frac{text{F}}{text{m}}) (farads per meter) |
Conversions
One Tesla equals one weber (the SI unit of magnetic flux) per square meter:
One ampere equals one coulomb (the SI unit of electric charge) per second:
Discovers of the law¶
The first observation that spurred researchers to look for the relationshiplinking magnetic field and current was made by Hans Christian Ørsted in 1820,who noticed that magnetic needles were deflected by electric currents. Thisled several physicists in Europe to study this phenomenon in parallel. WhileJean-Baptiste Biot and Félix Savart were experimenting with a setup similar toØrsted's experiment (that lead them to define in 1820 a relationship known nowas the Biot-Savart's law), André-Marie Ampère's experiment focused onmeasuring the forces that two electric wires exert on each other. Heformulated the Ampere's circuital law in 1826 [Gri99], whichrelates the magnetic field associated with a closed loop to the electriccurrent passing through it. In its original form, the current enclosed by theloop only refers to free current caused by moving charges, causing severalissues regarding the conservation of electric charge and the propagation ofelectromagnetic energy.
In 1861 [Max61], James Clerk Maxwell extended Ampere's law by introducing thedisplacement current into the electric current term to satisfythe continuity equation of electric charge. Based on the idea of displacementcurrent, in 1864 [Max65], Maxwell established the theory of electromagneticfield, predicating the wave propagation of electromagnetic fields and theequivalence of light propagation and electromagnetic wave propagation.
It was not until the late 1880s [Her93], Heinrich Hertz experimentally proved the existenceof electromagnetic waves as predicated by Maxwell's electromagnetic theory, anddemonstrated the equivalence of electromagnetic waves and light.
These efforts have lain solid foundations for the development of modern electromagnetism.
Maxwell's equations are the basic equations of electromagnetism which are a collection of Gauss's law for electricity, Gauss's law for magnetism, Faraday's law of electromagnetic induction and Ampere's law for currents in conductors. Maxwell equations give a mathematical model for electric, optical, and radio technologies, like power generation, electric motors, wireless communication, radar, and, Lenses, etc. These Equations explain how magnetic and electric fields are produced from charges.
These equations are part of the comprehensive and symmetrical theory of electromagnetism, which is essential to understand electromagnetic waves, optics, radio and TV transmission, microwave ovens and magnetically levitated trains.
What are the four Maxwell's Equations?
The four of Maxwell's equations for free space are:
The First Maxwell's equation (Gauss's law for electricity)
The Gauss's law states that flux passing through any closed surface is equal to 1/ε0 times the total charge enclosed by that surface.
If we deal with linear isotropic media then we have
and the Ampere-Maxwell equations can be written as
Units¶
Magnetic B-field | (mathbf{b}) | T | tesla |
Electric field intensity | (mathbf{e}) | (frac{text{V}}{text{m}}) | volt per meter |
Electric current | (text{I}) | A | ampere |
Electric current density | (mathbf{j}) | (frac{text{A}}{text{m}^{2}}) | ampere per square meter |
Magnetization | (mathbf{m}) | (frac{text{A}}{text{m}}) | ampere per meter |
Electric polarization | (mathbf{p}) | (frac{text{A}cdottext{s}}{text{m}}) | ampere times seconds per square meter |
Magnetic H-field | (mathbf{h}) | (frac{text{A}}{text{m}}) | ampere per meter |
Electric displacement | (mathbf{d}) | (frac{text{C}}{text{m}^{2}}) | coulomb per square meter |
Constants
Magnetic constant | (mu_0 = 4pi ×10^{−7} frac{text{N}}{text{A}^2} approx 1.2566370614..×10^{-6} frac{text{T}cdot text{m}}{text{A}}) |
Vacuum permittivity | (varepsilon_0 approx 8.854 187 817.. × 10^{−12} frac{text{F}}{text{m}}) (farads per meter) |
Conversions
One Tesla equals one weber (the SI unit of magnetic flux) per square meter:
One ampere equals one coulomb (the SI unit of electric charge) per second:
Discovers of the law¶
The first observation that spurred researchers to look for the relationshiplinking magnetic field and current was made by Hans Christian Ørsted in 1820,who noticed that magnetic needles were deflected by electric currents. Thisled several physicists in Europe to study this phenomenon in parallel. WhileJean-Baptiste Biot and Félix Savart were experimenting with a setup similar toØrsted's experiment (that lead them to define in 1820 a relationship known nowas the Biot-Savart's law), André-Marie Ampère's experiment focused onmeasuring the forces that two electric wires exert on each other. Heformulated the Ampere's circuital law in 1826 [Gri99], whichrelates the magnetic field associated with a closed loop to the electriccurrent passing through it. In its original form, the current enclosed by theloop only refers to free current caused by moving charges, causing severalissues regarding the conservation of electric charge and the propagation ofelectromagnetic energy.
In 1861 [Max61], James Clerk Maxwell extended Ampere's law by introducing thedisplacement current into the electric current term to satisfythe continuity equation of electric charge. Based on the idea of displacementcurrent, in 1864 [Max65], Maxwell established the theory of electromagneticfield, predicating the wave propagation of electromagnetic fields and theequivalence of light propagation and electromagnetic wave propagation.
It was not until the late 1880s [Her93], Heinrich Hertz experimentally proved the existenceof electromagnetic waves as predicated by Maxwell's electromagnetic theory, anddemonstrated the equivalence of electromagnetic waves and light.
These efforts have lain solid foundations for the development of modern electromagnetism.
Maxwell's equations are the basic equations of electromagnetism which are a collection of Gauss's law for electricity, Gauss's law for magnetism, Faraday's law of electromagnetic induction and Ampere's law for currents in conductors. Maxwell equations give a mathematical model for electric, optical, and radio technologies, like power generation, electric motors, wireless communication, radar, and, Lenses, etc. These Equations explain how magnetic and electric fields are produced from charges.
These equations are part of the comprehensive and symmetrical theory of electromagnetism, which is essential to understand electromagnetic waves, optics, radio and TV transmission, microwave ovens and magnetically levitated trains.
What are the four Maxwell's Equations?
The four of Maxwell's equations for free space are:
The First Maxwell's equation (Gauss's law for electricity)
The Gauss's law states that flux passing through any closed surface is equal to 1/ε0 times the total charge enclosed by that surface.
Ampere's Force Law
Integral form of Maxwell's 1st equation
It is the integral form of Maxwell's 1st equation.
Ampere's Law In Differential Form
Maxwell's first equation in differential form
It is called the differential form of Maxwell's 1st equation.
The Second Maxwell's equation (Gauss's law for magnetism)
The Gauss's law for magnetism states that net flux of the magnetic field through a closed surface is zero because monopoles of a magnet do not exist.
The Third Maxwell's equation (Faraday's law of electromagnetic induction )
According to Faraday's law of electromagnetic induction
It is the differential form of Maxwell's third equation.
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The Fourth Maxwell's equation ( Ampere's law)
The magnitude of the magnetic field at any point is directly proportional to the strength of the current and inversely proportional to the distance of the point from the straight conductors is called Ampere's law.
Third Maxwell's equation says that a changing magnetic field produces an electric field. But there is no clue in fourth Maxwell's equation whether a changing electric field produces a magnetic field? To overcome this deficiency, Maxwell's argued that if a changing magnetic flux can produce an electric field then by symmetry there must exist a relation in which a changing electric field must produce a changing magnetic flux.
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