Maxwell's equations are used describe electricity and magnetism. The first two tell us about electric and magnetic effects, but the last two are the most important because they unite electricity and magnetism; hence, “electromagnetism". Their initial state are integrals in the form of differential equations in which you NEED to actually learn about how those animals are derived into standard form because they might not make any sense to you now. However, we can peek into what these equations mean. Differential equations break down into functions. They are basically relations where big functions spit out smaller functions and yield general solutions, and with a little more work, particular solutions. These relations basically state that moving charges generate magnetic fields and changing magnetic fields create changing electric fields.
A Dynamical Theory of the Electromagnetic Field is the third of James Clerk Maxwell's papers regarding electromagnetism, published in 1865. It is the paper in which the original set of four Maxwell's equations first appeared. The concept of displacement current, which he had introduced in his 1861 paper On Physical Lines of Force, was utilized for the first time, to derive the electromagnetic wave equation. But now for the details…
The First Equation
Gauss's Law of Electricity is the first equation which describes electric flux in an enclosed space. Flux is the electric flow rate which is directly proportional to its number of electric field lines and the charge. So, if you have a point-charge in space, Coulomb's law accommodates this equation since it deals with forces exerted on a charge. E indicates a vector in any space of an electric field which is equivalent to the point charge (q) over epsilon (permittivity in space) which is then equivalent to the surface area of a sphere times the constants k and q. The integral sign basically tells us that the field is infinite in an enclosed space. This is where we have a Gaussian surface. Gaussian surfaces are how we measure effects in an enclosed surface. It's like taking a slice of the pie and measuring its contents to get an approximation from any chosen parameters– hence where integration limits come from.
The Second Equation
The second equation In the second equation we see is the law of magnetism. This says that magnetic flux is the field perpendicular multiplied together to the area and it is always equal to zero in an enclosed space. So, B in this case is the magnetic field in a Gaussian area. If it is non-zero, we have a problem. Then we'd have magnetic monopoles which *cannot exist. Even more simply put - you cannot have a magnet with only an N side or an S side. Dipoles only (N and S)!
You can do this experiment at home. If you cut a conducting metal in half what happens? Put some iron filings or something attractive to the metal. Does the north pole stay north and the south pole stay south?
The Third Equation
The third equation is Faraday's law. AKA: The law of induction. You may have heard of inductors, which are coils/loops that control the flow of current. It’s a current regulator of sorts. It resists change. This results in emf (electromotive force or induced voltage). To get an emf, the field must be increased OR its area or orientation must be changed. A potential difference (work done per unit charge if you are familiar with Ohm’s law and resistance) between two points across a conductor in varying fields allows us to predict how the magnetic field will interact with an electric circuit producing emf. So what this equation is basically saying is the electric field around a closed loop is proportional to the magnetic flux in a closed loop with respect to time.
The Fourth Equation
Now here we have the final equation – Ampere’s Law. When there is no moving field, the path integral of the magnetic field around a closed loop is proportional to the electric current that goes through the loop. Notice the little circle around the second integral sign is gone! This tells us that conservation of charge happens. Conservation of charge states that current going through an enclosed Gaussian surface is proportional to the charge within the surface with respect to time. Those small deltas you see (δ) show an infinitesimally small change over time. Mu (µ) is the permeability constant (free space) taken from the Biot-Savart Law and “i” refers to current.
The Second Equation
The second equation In the second equation we see is the law of magnetism. This says that magnetic flux is the field perpendicular multiplied together to the area and it is always equal to zero in an enclosed space. So, B in this case is the magnetic field in a Gaussian area. If it is non-zero, we have a problem. Then we'd have magnetic monopoles which *cannot exist. Even more simply put - you cannot have a magnet with only an N side or an S side. Dipoles only (N and S)!
You can do this experiment at home. If you cut a conducting metal in half what happens? Put some iron filings or something attractive to the metal. Does the north pole stay north and the south pole stay south?
The Third Equation
The third equation is Faraday's law. AKA: The law of induction. You may have heard of inductors, which are coils/loops that control the flow of current. It’s a current regulator of sorts. It resists change. This results in emf (electromotive force or induced voltage). To get an emf, the field must be increased OR its area or orientation must be changed. A potential difference (work done per unit charge if you are familiar with Ohm’s law and resistance) between two points across a conductor in varying fields allows us to predict how the magnetic field will interact with an electric circuit producing emf. So what this equation is basically saying is the electric field around a closed loop is proportional to the magnetic flux in a closed loop with respect to time.
The Fourth Equation
Now here we have the final equation – Ampere’s Law. When there is no moving field, the path integral of the magnetic field around a closed loop is proportional to the electric current that goes through the loop. Notice the little circle around the second integral sign is gone! This tells us that conservation of charge happens. Conservation of charge states that current going through an enclosed Gaussian surface is proportional to the charge within the surface with respect to time. Those small deltas you see (δ) show an infinitesimally small change over time. Mu (µ) is the permeability constant (free space) taken from the Biot-Savart Law and “i” refers to current.
Questions: Did you notice the similarities between electricity and magnetism in the third and fourth equations? Did you also see how the first two equations are not time-dependent? Why do you think that is?
Postscript :
* In classical theory magnetic monopoles are forbidden, but in QED (Quantum Electrodynamics) there are Dirac Strings which can be treated as magnetic monopoles. Dirac was the first guy who wrote the solution for the magnetic monopole. He realized that the vector potential couldn't be defined globally because the divergence of "B" is no longer quite zero, but a Dirac delta function. But one could define the vector potential almost globally - in the whole space with a semi-infinite Dirac string removed. A Dirac string is a thin solenoid that connects the magnetic monopole with the opposite monopole at infinity. Together, they make up an ordinary magnet with both poles. The other monopole at infinity is undetectable because of the distance but the solenoid is only invisible if the Aharonov-Bohm effect produces no phase for charged particles orbiting the solenoid. That led Dirac to realize the Dirac quantization rule - the fact that the magnetic monopoles' charges must be multiples of "2π/e", where "e" is the minimum elementary electric charge. Dirac's paper is really fascinating and a must read.
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