Maxwell's equations are a set of four equations that form the foundation of classical electromagnetism, optics, and electric and magnetic circuits. They explain the fundamentals of electricity and magnetism and are used to model electromagnetic waves, also known as light.
The equations describe how electric and magnetic fields are generated by charges, currents, and changes in the fields. They can be written in two ways: as integrals or as spatial derivatives.
The four equations are:
1. Electric field diverges from electric charge, an expression of the Coulomb force.
2. There are no isolated magnetic poles, but the Coulomb force acts between the poles of a magnet.
3. Electric fields are produced by changing magnetic fields, an expression of Faraday's law of induction.
4. Circulating magnetic fields are produced by changing electric fields and by electric currents, Maxwell's extension of Ampère's law to include the interaction of changing fields.
These equations have been used to develop a wide range of modern technologies, including electric cars, phones, electric motors, and the Large Hadron Collider. They have also been used to deepen our understanding of the universe, such as by allowing us to observe the universe through different portions of the electromagnetic spectrum.
Characteristics | Values |
---|---|
Number of equations | 4 |
First equation | Electric field diverges from electric charge, an expression of the Coulomb force |
Second equation | There are no isolated magnetic poles, but the Coulomb force acts between the poles of a magnet |
Third equation | Electric fields are produced by changing magnetic fields, an expression of Faraday's law of induction |
Fourth equation | Circulating magnetic fields are produced by changing electric fields and by electric currents, Maxwell's extension of Ampère's law to include the interaction of changing fields |
What You'll Learn
Electric field diverges from electric charge
Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric and magnetic circuits. The equations describe how electric and magnetic fields are generated by charges, currents, and changes in the fields.
In a four-dimensional world, Maxwell's equations can be generalised using the field tensor Fμν. In a 3+1D space, this tensor looks like:
! [F tensor](https://i.stack.imgur.com/d0j1K.png)
If you apply a rotation matrix corresponding to a 3D spatial rotation to this tensor, you will find that it mixes up the three components E1, E2, and E3, and it separately mixes up the three components B12, B13, and B23. The way in which these components mix is consistent with E being a vector in 3D space and with B being a separate vector in 3D space.
However, if you apply a Lorentz boost (a change in velocity), which is just a rotation that incorporates the time dimension as well as the three spatial dimensions, you will find that it mixes the components of E with the components of B. Nothing in the theory of normal 3D vectors allows this to happen. So if you didn't know about F, this would be your first clue that E and B are not really vectors, but instead are part of some more complex structure.
Anyway, once you know what F really looks like, it's straightforward to generalise it to higher dimensions. In 4+1D, for example, the tensor Fμν looks like:
! [F tensor in 4+1D](https://i.stack.imgur.com/nU60t.png)
This has four elements of an electric nature, but six of a magnetic nature. If you apply a 4D spatial rotation matrix to this, you will still find that the four components E1, E2, E3, and E4 mix in the way you expect of a 4D vector, but the six magnetic components don't. So if we lived in a 4+1D universe, it would be obvious that the magnetic field is not a vector, if nothing else because it has more elements than a 4+1D spatial vector does.
Maxwell's equations can be expressed using the cross product in three dimensions using the antisymmetric tensor εαβγ. The cross product can be written as:
! [Cross product](https://i.stack.imgur.com/bJ4nT.png)
However, the cross product and curl are only defined for three spatial directions. To generalise Maxwell's equations to higher dimensions, we can use the mathematical machinery of exterior differential forms, exterior wedge product (∧), and exterior de Rham differential (d).
In a four-dimensional world, the number of electric fields is three, the number of magnetic fields is six, and there are eight Maxwell equations.
The divergence of the electric field describes the conservation of electric charge. Gauss' law is a constraint, and any solution of Maxwell's equations satisfies it. An electric field points away from positive charges and towards negative charges, and the net outflow of the electric field through a closed surface is proportional to the enclosed charge, including bound charge due to polarisation of material.
In conclusion, the electric field diverges from electric charge.
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There are no isolated magnetic poles
Gauss's law for magnetism states that there are no isolated magnetic poles. In other words, there are no magnetic monopoles. This is in contrast to electric charges, which can exist in isolation as positive or negative poles.
The law states that the closed surface integral of magnetic flux density is always equal to the total scalar magnetic flux enclosed within that surface, regardless of its shape, size, or the medium in which it lies. Mathematically, this can be expressed as:
> ∮ B · ds = 0
Here, B represents the magnetic field. Applying the Gauss divergence theorem, this can be converted into a volume integral by taking the divergence of the same vector:
> ∮ B · ds = ∭ ∇ · B dv
Substituting the equation above, we get:
> ∭ ∇ · B dv = 0
To satisfy this equation, either the volume must be zero or the divergence of the magnetic field must be zero. Since the volume of any object or body cannot be zero, we arrive at the conclusion that the divergence of the magnetic field is zero:
> ∇ · B = 0
This means that the net outflow of the magnetic field through a closed surface is zero, and magnetic field lines always form closed loops.
The absence of isolated magnetic poles has important implications. Unlike electric fields, which can be created by electric charges, magnetic fields cannot be produced by magnetic monopoles. Instead, magnetic fields arise from dipoles, where the north and south poles are inseparable. This fundamental difference between electric and magnetic fields has significant consequences in the behaviour of electromagnetic waves and the nature of electromagnetic interactions.
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Electric fields are produced by changing magnetic fields
Maxwell's equations, which form the foundation of classical electromagnetism, describe how electric and magnetic fields are generated by charges, currents, and changes in the fields. One of these equations, known as Faraday's Law, describes how a changing magnetic field induces an electric field.
Electric Fields Produced by Changing Magnetic Fields
The relationship between changing magnetic fields and the resulting electric fields is described by Faraday's Law, which is one of Maxwell's four equations. Faraday's Law states that a time-varying magnetic field will always produce an electric field.
In mathematical terms, Faraday's Law can be expressed as:
> emf_alt = -dφ/dt
Where emf_alt is the alternating electromotive force induced in a coil and φ is the scalar magnetic flux. The negative sign indicates that the induced electromotive force always opposes the time-varying magnetic flux, as described by Lenz's Law.
This equation can be rewritten in terms of the magnetic field B and the electric field E as:
> ∫ E ⋅ dl = -d/dt ∫ B ⋅ dA
This equation states that the closed line integral of the electric field E is equal to the negative rate of change of the surface integral of the magnetic field B.
Using Stokes' theorem, this equation can be further simplified to:
> ∫ (∇ × E) ⋅ dA = -d/dt ∫ B ⋅ dA
Finally, by cancelling the surface integral on both sides, we arrive at the compact form of Faraday's Law:
> ∇ × E = -dB/dt
This equation shows that the curl of the electric field E is equal to the negative rate of change of the magnetic field B.
Examples and Applications
The principle described by Faraday's Law has numerous practical applications and is used in various devices and technologies.
One simple example is a magnetic compass placed next to a wire carrying an electric current. When current passes through the wire, the compass needle will deflect, indicating the presence of a magnetic field circling the wire. This phenomenon was first observed by Hans Christian Ørsted in 1820 and established the close relationship between electricity and magnetism.
Faraday's Law is also the principle behind electric generators and motors. An electric generator works by rotating a coil of wire inside a magnetic field, inducing an electric current in the wire. An electric motor works on the same principle but in reverse, using an electric current to generate a magnetic field that interacts with a stationary magnetic field to produce motion.
Maxwell's Equations in Higher Dimensions
While Maxwell's equations are typically written and applied in three spatial dimensions, they can be generalized to higher dimensions. In four spatial dimensions, for example, the electric field E would have four components, while the magnetic field B would have six components.
The mathematical formulation becomes more complex in higher dimensions, and the familiar vector notation used in three dimensions is no longer sufficient. Instead, the equations are expressed using tensor notation, which allows for the representation of objects with more than three dimensions.
In summary, Maxwell's equations, including Faraday's Law, describe how changing magnetic fields induce electric fields. This principle has important practical applications in various technologies and can be generalized to higher-dimensional spaces using tensor mathematics.
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Circulating magnetic fields are produced by changing electric fields and by electric currents
Maxwell's equations describe how electric and magnetic fields are generated by charges, currents, and changes in the fields. The equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, electric circuits, and classical optics.
The Ampère-Maxwell law states that magnetic fields relate to electric currents. Maxwell's addition to Ampère's circuital law states that magnetic fields also relate to changing electric fields, which he called the displacement current. The integral form of the Ampère-Maxwell law states that electric and displacement currents are associated with a proportional magnetic field along any enclosing curve.
The time-varying magnetic field will always produce an electric field. This is derived from Faraday's laws of electromagnetic induction, which state that whenever there are N-turns of a conducting coil in a closed path placed in a time-varying magnetic field, an alternating electromotive force is induced in each coil.
The magnetic field is a physical field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and to the magnetic field.
A magnetic field describes a volume of space where there is a change in energy. The motion of an electric charge producing a magnetic field is an essential concept in understanding magnetism. The strength of the magnetic field is called the magnetic moment.
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The equations reveal a whole spectrum of invisible waves
The electromagnetic spectrum is the full range of electromagnetic radiation, which can be organized by frequency or wavelength. The spectrum is divided into separate bands, with different names for the electromagnetic waves within each band. From low to high frequency, these are: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays.
The equations of James Clerk Maxwell, known as Maxwell's equations, form the foundation of classical electromagnetism, classical optics, and electric and magnetic circuits. They describe how electric and magnetic fields are generated by charges, currents, and changes in the fields.
Radio waves, at the low-frequency end of the spectrum, have the lowest photon energy and the longest wavelengths—thousands of kilometers or more. They can be emitted and received by antennas and pass through the atmosphere, foliage, and most building materials.
Gamma rays, at the high-frequency end of the spectrum, have the highest photon energies and the shortest wavelengths—much smaller than an atomic nucleus. Gamma rays, X-rays, and extreme ultraviolet rays are called ionizing radiation because their high photon energy allows them to ionize atoms, causing chemical reactions.
Visible light, the light we can see, occupies a narrow band in the broad range of types of electromagnetic radiation. It is a combination of the colors of the rainbow, from lower-frequency red light to higher-frequency violet light.
Ultraviolet radiation, or UV, sits between the violet end of the visible spectrum and the X-ray range. UV rays can ionize atoms, causing chemical reactions, and can also break chemical bonds, making molecules unusually reactive.
X-rays are highly penetrating and can pass through many substances with little absorption. They are used in diagnostic imaging in medicine, such as radiography, and in astronomy to study the accretion disks around neutron stars and black holes.
Gamma rays are the most energetic photons, with no defined lower limit to their wavelength. They are used in medical imaging, such as PET scans, and in cancer therapy.
The different types of electromagnetic radiation are broadly classified by their interactions with matter, which vary across the spectrum.
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Frequently asked questions
Maxwell's laws, or Maxwell's equations, are a set of four equations that form the foundation of classical electromagnetism, optics, and electric and magnetic circuits. They describe how electric and magnetic fields are generated by charges, currents, and changes in the fields.
In a four-dimensional world, Maxwell's laws are generalised using the field tensor Fμν. The number of electric and magnetic field components, as well as the number of Maxwell's equations, changes depending on the number of spacetime dimensions. In four spacetime dimensions, there are four electric field components, six magnetic field components, and eight Maxwell's equations.
Maxwell's laws have had a profound impact on modern technology. They have enabled virtually all modern electrical, electronic, and photonic technologies. They also led to the discovery of electromagnetic waves, including radio waves and microwaves, which are used in wireless communication, radar, and satellite communication.