Maxwell's equations

In this blog/article, I demonstrate bringing Maxwell's equations represent one of the most elegant and concise ways to state the fundamentals of electricity and magnetism. From them one can develop most of the working relationships in the field. Because of their concise statement, they embody a high level of mathematical sophistication and are therefore not generally introduced in an introductory treatment of the subject, except perhaps as summary relationships.

...James Clerk Maxwell ...

Maxwell's Equations :

Maxwell’s four equations describe the electric and magnetic fields arising from distributions of electric charges and currents, and how those fields change in time.   They were the mathematical distillation of decades of experimental observations of the electric and magnetic effects of charges and currents, plus the profound intuition of Michael Faraday.  Maxwell’s own contribution to these equations is just the last term of the last equation—but the addition of that term had dramatic consequences.  It made evident for the first time that varying electric and magnetic fields could feed off each other—these fields could propagate indefinitely through space, far from the varying charges and currents where they originated.  Previously these fields had been envisioned as tethered to the charges and currents giving rise to them.   Maxwell’s new term (called the displacement current) freed them to move through space in a self-sustaining fashion, and even predicted their velocity—it was the velocity of light!

Maxwell First Equation :

Maxwell first equation is based on the Gauss law of electrostatic which states that “when a closed surface integral of electric flux density is always equal to charge enclosed over that surface”
Mathematically Gauss law can be expressed as,
Over a closed surface the product of electric flux density vector and surface integral is equal to the charge enclosed.

∯D⃗ .ds¯=Qenclosed      ——(1)

Any closed system will have multiple surfaces but a single volume. Thus, the above surface integral can be converted into a volume integral by taking the divergence of the same vector. Thus, mathematically it is-

∯D¯.ds¯=∭▽.D⃗ dv⃗  —-(2)

Thus, combining (1) and (2) we get-

∭.D . dv =Qenclosed —–(3)

Charges in a closed surface will be distributed over its volume. Thus, the volume charge density can be defined as –

 ρv=dQdv  measured using C/m3

 On rearranging we get-

 dQ=ρvdv

 On integrating the above equation we get-

 Q=∭ρvdv ——-(4)

 The charge enclosed within a closed surface is given by volume charge density over that volume.

 Substituting (4) in (3) we get-

 ∭▽.Ddv=∭ρvdv

 Canceling the volume integral on both the sides, we arrive at Maxwell’s First Equation-

=▽.Ddv=ρv

Maxwell's second equation  :Gauss law of magnetism 

The corresponding formula for magnetic fields:

∫B→⋅dA→=0.$$

(No magnetic charge exists: no “monopoles”.)

3. Faraday’s Law of Magnetic Induction: 

∮E→⋅dℓ→=−d/dt(∫B→⋅dA→).$$ 

The first term is integrated round a closed line, usually a wire, and gives the total voltage change around the circuit, which is generated by a varying magnetic field threading through the circuit.

4.Ampere’s Law plus Maxwell’s displacement current: 

∮B→⋅dℓ→=μ0(I+ddt(ε0∫E→⋅dA→)).$$ 

This gives the total magnetic force around a circuit in terms of the current through the circuit, plus any varying electric field through the circuit (that’s the “displacement current”).

The purpose of this lecture is to review the first three equations and the original Ampere’s law fairly briefly, as they were covered earlier in the course, then to demonstrate why the displacement current term must be added for consistency, and finally to show, without using differential equations, how measured values of static electrical and magnetic attraction are sufficient to determine the speed of light.

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