Importance of Maxwell’s Equations.
Brilliant Electromagnetic Synthesis of Electrical & Magnetic Effects. Maxwell’s equations synthesized physicists’ understanding of electrodynamics and predicted the existence of electromagnetic waves.
By the middle of the 19th century physicists had deduced, in bits and pieces, that electrical and magnetic phenomena were connected. They had learned some of the details of electrodynamics, but did not yet understand how everything fit together. After over a decade of working on the problem, Scottish physicist, James Clerk Maxwell, published papers in 1865 and 1868 that synthesized all electromagnetic phenomena into four equations. Maxwell then used these equations to predict the existence of electromagnetic waves and demonstrate that light was an electromagnetic wave.
Maxwell’s first equation is Gauss’s law. The electric flux through a surface area is what physicists use to measure the total amount of electric field going through that surface. Physically this equation says that the total electric flux integrated over a surface that encloses a volume of space is proportional to the total charge contained in that volume of space. Furthermore this equation implies that electric charges create electric fields.
Gauss’s Law for Magnetism
Maxwell’s second equation is Gauss’s Law for Magnetism. Notice the similarities and differences between the first and second equations. The left hand side of the first equation is the total electric flux while in the second equation it is the total magnetic flux. The major difference, however, is in the right hand side of the equations. The electric version of Gauss’s law has the total charge enclosed, while in the magnetic version the total integrated magnetic flux always equals zero.
It seems that if electrical and magnetic phenomena behaved analogously, then the total magnetic flux would be proportional to the magnetic charge enclosed. Well it is, but the total magnetic charge enclosed must be zero. Always! Maxwell’s second equation means physically that there can be no isolated magnetic poles, which are called magnetic monopoles. Magnetic monopoles are the magnetic equivalent of isolated positive or negative electric charges. North and south magnetic poles must always exist in pairs. Isolated north or south magnetic poles do not exist.
Maxwell’s third equation is Faraday’s Law. The equation relates a changing magnetic flux to an electric field. Physically this means that when a magnetic field changes in some way it creates an electric field. Electric fields can result either from electric charges or changing magnetic fields.
Maxwell’s Extension of Amperes Law
Maxwell’s fourth equation is an extension of Ampere’s law. Maxwell added another term to the right hand side of the equation for Ampere’s law that is analogous to the right hand side of Faraday’s law. Where Faraday’s law has the derivative of the magnetic flux, Maxwell’s new term in this equation has the derivative of the electric flux. Physically this term means that a changing electric field can create a magnetic field. In fact because no magnetic monopoles exist all magnetic fields are created by some type of changing electric field.
Notice that in Faraday’s law Maxwell did not add a magnetic current term similar to the electric current term in Ampere’s law. That is because without magnetic monopoles there can be no magnetic current.
After synthesizing electromagnetic phenomena, Maxwell mathematically manipulated his equations to derive a wave equation. This electromagnetic wave equation correctly predicted the existence of electromagnetic waves. Furthermore the theoretical speed of these electromagnetic waves equaled the experimental values of the speed of light. Maxwell thereby demonstrated that light was a type of electromagnetic wave and synthesized electrodynamics and optics.
Maxwell’s equations rank along with Newton’s laws as among the most brilliant syntheses in classical physics.