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"Everything has got an equation"


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Maxwell equations explained

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Fields and waves

Fields and waves are important concepts in both physics and engineering. In physics, field is a physical quantity that has a value or a set of values at each point of space or space-time. So field is a function of coordinates. Scalar field is single-valued function and vector field is multi-valued function. For example we can think of gravitational field which exerts a force on every object placed in space. This is valid in newton's framework and the field has a single value at each point of space. In General relativity , gravitational field at each point in space-time has ten quantities.This gravitational field is called metric tensor. The apparent equivalence between metric tensor and gravitaional field had brought a revolutioin in physics, which we will discuss when explaining theory of general relativity. Electric field and magnetic field are vector quantities themselves. Electric field is the rate of change of potential.
maxwell equations

Electric field is the force which a unit change experiences. It can be defined using electric force lines that permeate some region of space. Electromagnetic field is a combination of electric field and magnetic field. In fact it is the cross product of those . So the direction of propagation is transverse to both. Waves are disturbances in some medium, which carry energy through that medium. Sound and light are transmitted as a form of wave. Sound wave is a longitudinal wave whereas light is a transverse wave. In case of longitudinal wave the direction of the propagation is parallel to the vibration in the medium. In case of transverse wave the direction of propagation is perpendicular to the vibration in the medium. Propagation of light is transverse to the plane of electric and magnetic field. Electric field creates magnetic field and pushes it. Magnetic field, in turns, creates electric field and pushes it. By their mutual interaction a disembodied wave is formed and it travels at the speed of light. For mathematical analysis , we will discuss the electromagnetic wave theory which was developed by James Clerk Maxwell . His theory of electromagnetism revolutionized the field of physics and engineering.The four maxwell's equations related to electric and magnetic field are

maxwell equations explained

Maxwell's equations are a perfect embodiment (example) of differential laws. Differential laws are very powerful and essential tools for describing continuous processes that occur in nature. Differential laws must be described by differential equation. In quantum mechanics where the processes are discrete such laws seems scarcely satisfactory.
The four equations ( in either differential or integral form) are coupled by the electric field and magnetic field (E, B). So They are not independent as changing electric and magnetic field are dependable with each other.

Divergence and Curl

In the figure above maxwell's equations are written in both differential an integral form. Differential form and integral form are known as microscopic and macroscopic form also. There are four equations that perfectly describes the nature of electromagnetic waves and their propagation. Four maxwell's equations must not be thought independent rather they are coupled. The first equation is basically the Gauss's law of electric charge. It says that the total outward flux throughout a closed surface is equal to the total charge enclosed by the surface multiplied by a constant. Flux is any physical quantity that flows through a medium or surface and is defined as the product of any field and component of an area that the field intersect perpendicularly. As an example, the electrical flux is the electric field multiplied by a small planar area perpendicular to field lines. When the area is not planar then evaluation of the flux requires area or surface integral to be taken over the area as the angle will be always changing. Some examples of flux are electric and magnetic flux. Flux can be vector quantity also. And example of flux integral can be seen when we derive Gauss's law from Culomb's law

Gauss's law

Some charge q is assumed to exist at a point at space and we want to find the flux that it creates around it. So we can imagine a spherical surface around the charge and take surface integral. The surface is curved (Gaussian) so we are bound to take surface integral. As the angle is zero between infinitesimal area ds and E , the surface integral get simplified as given in the second equation. It is just the integral of (Ecos(0)dA= EdA). Finally we can take E out of integral as E does not depends on ds. We know that total surface area of sphere of radius r is 4(3.14)r(squared). Thus we can arrive at the equation for flux , which corresponds to Gauss's original law.

Definitions of μ0 and ε0

Ampere uncovered that two parallel wires carrying electric currents in the same direction attract each other magnetically, the force in newtons per unit length being given by F=2(μ04π)(I1)(I2)/r,
for long wires a distance r apart. We are using the standard modern units (SI). The constant μ0/4π that appears here is exactly 10^(-7), this defines our present unit of current, the ampere(A). To repeat: μ0/4π is not something to measure experimentally, it's just a funny way of writing the number 10^(-7) That's not quite fair ; it has dimensions to ensure that both sides of the above equation have the same dimensionality. (Of course, there's a historical reason for this strange convention, as we shall see later). Anyway, if we bear in mind that dimensions have been taken care of, and just write the equation
F= 2X10^(-7)I1XI2/r
it's clear that this defines the unit current(one ampere) as that current in a long straight wire which exerts a magnetic force of 2X10^(−7) newtons per meter of wire on a parallel wire one meter away carrying the same current.


The second maxwell's equation says that changing magnetic field creates electric field. If the magnetic field were not changing the electric field would not have been created. The differentiation of a constant function is zero. But this formula tells something more: the curl of electric field is the time rate of change of magnetic field.
The third maxwell's equation relates magnetic field with its divergence. It is apparent from the equation that the divergence of the magnetic field is zero. From this we can infer there is no isolated magnetic mono pole. The magnetic field lines always close on in itself. This is hard to explain without mathematics but it is apparent that some properties are fundamentally different in case of magnetic field.

The fourth formula is actually the modified version of Ampere's circuit law. Maxwell added an extra term to accommodate for the displacement current, which is the integral of time rate of change of electric field, multiplied by constant term. We can visualize this displacement current total charge that is deposited on the capacitor plate. It is not actual current but kind of electric field as no current flows through two capacitor plates.


Problem with Ampere's Law

A simple example to see that Ampere's Law must be faulty in the general case is given by Feynman in his Lectures in Physics. Suppose we use a hypodermic needle to be inserted a spherically symmetric blob of charge in the middle of a large vat of solidified jello (which we assume conducts electricity). Because of electrostatic repulsion, the charge will dissipate, currents will flow outwards in a spherically symmetric fasion. Question: does this outward-moving current distribution create a magnetic field? The answer must be no , because since we have a completely spherically symmetric case, it could only generate a spherically symmetric magnetic field. But the only possible such fields are one pointing outwards everywhere and one pointing inwards everywhere, both corresponding to non-existent monopoles. So, there can be no magnetic field.

ampere circuit law

However, imagine we now consider checking Ampere's law by taking as a path a horizontal circle with its center above the point where we injected the charge (think of a halo above someone’s head.) Surely, the left hand side of Ampere's equation is zero, since there can be no magnetic field's existence. (It would have to be spherically symmetric, meaning radial.) On the other hand, the right hand side is most definitely not zero, since some of the outward flowing current is going to go through our circle. So the equation must be erroneous.
Ampere's law was established as the result of large numbers of careful experiments on all kinds of current distributions. So how could it be that something of the kind we discussed above was overlooked? The reason is really similar to why electromagnetic induction was missed for so long. No-one thought about looking at dynamic fields, all the experiments were done on steady situations. With our ball of charge spreading outward in the jello, there is surely a changing electric field. Imagine yourself in the jello near where the charge was injected: at first, you would feel a strong field from the nearby concentrated charge, but as the charge spreads out spherically, some of it going past you, the field will weaken with time.

Maxwell's reasoning

Maxwell himself gave a more practical example: consider Ampere's law for the usual infinitely long wire carrying a steady current I, but now break the wire at some point and put in two large circular metal plates, a capacitor, maintaining the steady current I in the wire everywhere else, so that charge is simply accumulating on one of the plates and draining off the other.
Looking now at the wire some distance away from the plates, the situation appears normal, and if we put the usual circular path around the wire, application of Ampere's law tells us that the magnetic field at distance r, from


ampere circuit law

Recall, however, that we defined the current threading the path in terms of current punching through a soap film spanning the path, and said this was independent of whether the soap film was flat, bulging out on one side, or whatever. With a single infinite wire, there was no escape— no twisting of this covering surface could wriggle free of the wire going through it (actually, if you distort the surface enough, the wire could penetrate it several times, but you have to count the net flow across the surface, and the new penetrations would come in pairs with the current crossing the surface in opposite directions, so they would cancel).


ampere circuit law

Once we bring in Maxwell's parallel plate capacitor, however, there is a way to deform the surface so that no current penetrates it at all: we can run it between the plates!
The question then arises: can we salvage Ampere's law by adding another term just as the electrostatic version of the third equation was rescued by adding Faraday's induction term? The answer is of course yes: although there is no current crossing the surface if we put it between the capacitor plates, there is definitely a changing electric field , because the capacitor is charging up as the current I flows in. Assuming the plates are close together, we can take all the electric field lines from the charge q on one plate to flow across to the other plate, so the total electric flux across the surface between the plates,


ampere circuit law
We finally arrive at maxwell's fourth equation.

Finally putting everything together we get :

maxwell equation

An analysis or study of vector calculus is necessary for the full treatment of Maxwell's equations. The divergence and curl used in the equations are simply measure of spacial variation of vector field. Divergence finds any source or sink from which something flows out. Divergence is calculated by finding dot product of any vector with an operator usually denoted by del(∇). So it is a scalar quantity. It can be expressed in any coordinate system and in Cartesian coordinate it has the simplest expression. In similar way curl operator can be expressed in any coordinate system. All are equivalent. Curl operation on vector field gives the infinitesimal rotation of that vector field in three dimensional euclidean space. Thus the curl of any vector is also a vector quantity.

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Consequences

The coupling of those equations gives rise to wave equation. The wave represented by the wave equation is the electromagnetic wave which is what we know as light or other electromagnetic radiation. Mathematician Jean-Baptiste le Rond d'Alembert first found the solution to wave equation. The wave equation is also called thed'Alembert's equation. That the solution to Maxwell's equations is electromagnetic wave should make sense because a wave is itself a function. We always expect solution to differential equation to be some kind of function. This wave is a function of space and time.

wave equation in 3D
Fig: wave equation

We first , as given in the figure, take curl of Maxwell's second equation. In the second equation we substitute B by H with a constant. Then using slightly modified form of Maxwell's fourth equation we get the penultimate(just before the last ) equation. Assuming constant sigma(σ) to be zero in free space we get the last equation which is the wave equation for electromagnetic radiation. The last equation is the wave equation for electric field. Similarly we can derive another wave equation for magnetic field. This is the wave equation in three dimensional euclidean space. We can find out the speed of the wave from the very equation. This wave equation is the standard form of all kind of transverse waves that exist in the physical world. Later we will use the same equation describing water waves, gravitational waves and others.

Wave is unchanging shape that travel across space. To be theoretic, wave is a function of space and time. But this is not to be confused with four dimensional space-time which is a continuum of events. The four equation of Maxwell are partial differential equations and they are linear. More discussion about the partial differential equation and their classifications deserve another mathematical treatise(essay). The physical world required for electromagnetic theory is three-dimensional but the theory is consistent with the theory of relativity . The four coupled equations of Maxwell can be written in tonsorial form which reduces the four equation to just two. electromagnetic field tensor captures all the information about the electromagnetic force. One of the two tensor equations says something like this:

Divergence of four dimensional electromagnetic field tensor in empty space is zero. ( the value of J is zero is empty space or vacuum).
There is a maxim like this : "the nature lay hidden and God says let there be light. After that there was light".
electromagnetic field tensor

The last equation in the figure above, is the divergence of electromagnetic field tensor, which is identically zero in empty space. Although the derivation is cumbersome but this is the compact and simplified version of Maxwell's equations. The equation is derived using Euler-Langrange equation which involves Langrangian (L). The Langrangian for electromagnetic field is given in the first equation. Lagrangian of a system is difference of the kinetic energy and potential energy. A system of equations involving Lagrangian is a more elegant and rigorous formulation of classical mechanics. It is widely known as Lagrangian mechanics. Other tensor equation of electromagnetism is :



electromagnetic field tensor

Summation of all covariant derivatives of electromagnetic field tensor is zero.

Tensor is a geometric objects that shows relationship between coordinate frames , coordinates and a transformation rule between them. Vectors and scalars are also tensors. Scalars are tensors of rank zero and vectors are tensors of rank one. Before the advent of theory of relativity space and time were considered independent and absolute. Classical world is four dimensional but not the one needed for electromagnetism. The theory of relativity is still a classical theory whereas quantum theory is a branch of modern physics. Although these theories are the two main strands of physics, electrical engineering can go without those as long as the application is concerned.

Maxwell's equation in other formalism

Maxwell's equations can be expressed in curved spacetime also. It can be expressed using differential forms and exterior algebra too. In vector calculus the fomulation is quite similar:
maxwell's equations

In Tensor calculus the equations are as follow :
maxwell's equations


maxwell's equations

In differential form the equations are :
maxwell's equations

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