Monday, March 23, 2015

Maxwell's Equaotions and c = 1/√ε0μ0


The degree of understanding of Maxwell's Equations (there are so many arguments on so called 'Maxwell's Equations') is quite up to you (or I would say how many times you have read the textbooks on the equations does relate little with your understanding not to mention how many textbooks you purchased or downloaded but how much you thought about them does). Basically the equations are the vector field equations (directions and intensities according to him and he presented many fields). The fundamental (though subtle to me) concept of the physics field itself differs from the Maxwell's field concept because the existence of 'ether' was experimentally denied after Maxwell. One remarkable thing done by Maxwell was that he predicted the following equation and the speed of light as transverse propagation of electromagnetic wave (undulations according to him) logically by using mathematics and the reports of the experiments done by some other physicists, which was and is regarded as a big discovery (or theoretical prediction), say once per two centuries (by British standard after Newton in 1600s). I think that the above underlined part cannot be neglected after reading his some original texts.






c = \frac{1}{\sqrt{ \mu_0 \varepsilon_0}} = 2.99792458 \times 10^8 \, \mathrm{m~s}^{-1}

The discovery of this equation was introduced in his On Physical Lines of Force (published in March, 1861). It is not a waste of time to check the original works to guess what really happened in the Maxwell's brain.

Maxwell was a specialist of capacitors and inductors as well as fluid mechanism. The title of Part One of his writing published in March, 1861 "On Physical Lines of Force" is "The Theory of Molecular Vortices applied to Magnetic". Vortices (Vortex in a singular form) is the key word and it is "Curl" or "Rotation" in Vector Calculus. As we will see later without double "Curl"operation we cannot reach a wave equation including 1/√ε0μ0. Diodes and transistors, even the ideas, did not exist in his time. He used the physics and electric terms which are different from those we use now but the fundamental things are largely still valid. And his insights and understandings were more fundamental due to his knowledge of mathematics and fluid mechanism.

We saw this equation in the last post " Sphere and Inverse-square law " and  μ0  is magnetic permeability in vacuum and  ε0 is electric permittivity in vacuum. In terms of units, μ0  is in H (Henry)/meter while ε0 is in F (Farad)/meter. We now recall the following basic four equations.


            dΦ              di(t)
v(t)  = ------  =  L -------   (LI = Φ)
             dt               dt

To make some converts

v(t)           L
-----   =   ----
 di(t)        d(t)


Unit wise

 v         L
---  =  ---
 i          t

We can connect this with Ohm's Law

V        
---  =  R
 I

Therefore

L = Rt

             dQ              dv(t)
i(t)  =  ------  =  C -------  (CV = Q)
             dt                dt


To make some converts

i(t)             C
-----   =   ----
dv(t)        d(t)

Unit wise

 I         C
---  =  ---
 V         t

We can connect this with Ohm's Law

  I          1         C
---  =  ----  =  -----
 V         R          t

Therefore

           t
C = ------
          R

Therefore

                    t
LC =  Rt x -----  = t2
                    R

This <unit wise> equation shows a very unique relation between L, C and time.


L = Inductance in Henry and C = Capacitance in Farad. Therefore

μ0  is in H (Henry) / meter = L (in H) / meter = V t (= Φ) / I x meter, or L (or mμ0 ) = Φ / I

ε0 is in F (Farad) / meter = C (in F) / meter = I t (= Q) / V x meter, or C (or mε0) = Q / V


Since

            dΦ    
v(t)  = -------   ----->  by integration with t,  Φ  =  vt
             dt

             dQ
i (t) =  --------  ----->  by integration with t,  Q  =  it
              dt

 Therefore Φ / I can be considered as
Φ / I  =  vt / i 

Divide both side by ' t ' (or more precisely differentiate with time)

Φ / I t = v / it  = V / Q (reciprocal of Q / V)
  
Apart from the (profound) meanings, unit wise μ0   =  1 / ε0 , or μ0  ε0 = 1. Is this correct ? Yes, unit wise.
The unit of μ0  is the same unit as that of 1 / ε0 .

while

mμ0    = L = Φ / I  =  vt / i 

μ0    =  vt / im

mε0  = C = Q / V =  it / v

ε0  =  it / vm

Therefore please note that μ0    and ε0 include the space dimensions, in this case 1-D, distance while the others v (V), i (I) , Q, Φ, C, L do not have dimensions.

μ0 ε0   , also unit wise =  ( vt / im) x (it / vm)  = vi t2 / viv m2 = t2 / m2 . Is this correct ? Yes, but not just unit wise but t2 / m2 (t / m)2 is the unit unit of μ0 ε0 .
                                                                     
Please recall LC Resonant Circuit equation
                ___
 f0 = 1 / i/L C 

               ___           ________               ____
f0 = 1 / i/L C = 1 / i/m μ0  m ε0  = 1 / m i/ε0 ε0


This has a meaning since

    1              m2
-------  =   ------
 μ0ε0               t2


    1                m
 -------  =    ------   =  velocity
   ____            t
 i/ε0 ε0  


And if we introduce distance or distance square, again unit wise we can get an acceleration and velocity.


Please note: 

Also  ＜Vdt ＞ is Φ (Magnetic Flux) and ＜Idt＞ is Q (Charge).

        dΦ
v = -------
         dt

        dQ
i = --------
         dt

vi = power

power x time = energy, so


 VIt  = VQ (Energy)   (Q in Electric Field, Potential energy)

C (or mε) = Q / V. This simple equation means:  C (Capacitor) keeps Q (Charge) against V (Voltage) and stores Energy in the form of VQ.


 IVt = IΦ (Energy) (I in Magnet Field, a kind of Dynamic energy)


L (mμ) = Φ / I. This simple equation means:  L (Inductor, Coil) creates and keeps Φ (Magnetic Fulx) against I (Electric Current) and stores Energy in the form of IΦ.

The key is <against>, which is similar to the Hooke's law of a spring, shows their inertial nature of C (ε0) and L (μ0), and a concept <restoring force (energy) >.

Precisely where and how to store energy in the form of VQ and IΦ is another story. An analogy of V to a spring is relatively easy to imagine. An analogy of I to a spring also seems to be easy when considering I (Electric Current) being a <current> or a <flow> or a <stream> of Q. But I (i) (Electric Current) is the time rate change of the amount of Q.

       dQ
i = -------
        dt

Not a <current> or a <flow> or a <stream> which has a velocity (v = distance / time). Q may move so it has a velocity. But in this case we must consider distance and time. The above equation simply shows the time rate change of the amount of Q (Electric Charges). How to count a number of Q ?


Hooke's law (from wiki)

"
Hooke's law for a spring is often stated under the convention that F is the restoring (reaction) force exerted by the spring on whatever is pulling its free end. In that case, the equation becomes
F= -k X\,
since the direction of the restoring force is opposite to that of the displacement.

"
In terms of Energy (from wiki)

"
The potential energy U_{{el}}(x) stored in a spring is given by
U_{{el}}(x)={1 \over 2}kx^{2}
which comes from adding up the energy it takes to incrementally compress (or extend) a spring. That is, the integral of force over displacement.

"
(underlined by me)

This reminds us of the stored energy in a Capacitor and an Inductor.

 1        2
---  Cv
 2 

CV = Q, so QV  is energy.


 1        2
---   Li
 2


LI = Φ , so ΦI is energy.

QVΦI = QΦVI = QΦP (VI = P, Power)


 Again the basic two equations

        dΦ
v = -------
         dt

When Φ is constant (no change of Φ with time),  v = 0.

        dQ
i = --------
         dt

When Q is constant (no change of Q with time),  i = 0.


To integrate the both side with ＜t＞(or 0 to ＜t＞)

vt = Φ

it = Q

So

                                                                                     
QVΦI = QΦVI = QΦP (VI = P: Power) = vtitP = ptPt = energy x energy (P x time = Energy).


Above we did QVΦI (QV x ΦI) but what is QVΦI (QV multiplied by ΦI) ?  Does this multiplication has any meaning ?  Also does the result "energy x energy" have any meaning ? Wrong !!, maybe.
QV x ΦI should be QV + ΦI (Addition).

QV seems a static energy derived from Q (fixed) multiplied by V (fixed). Verbally this static energy is proportional to the amount of Q and the strength (magnitude) of V.

On the other hand, ΦI is an Dynamic Energy and derived from Φ multiplied by I. This Dynamic energy is proportional to the amount of Φ and the strength (magnitude) of I. As the energy is dynamic, the amount of Φ changes with time and I changes with time, in which case denoted by a small letter <i>.

QV + Φi = ItV + Vti. Is this correct ?

 We must consider

 Q of QV is the amount of Q. However, QV = ItV is obtained by Q being integration with <t> by using

        dQ
i = --------
         dt


QV = ItV involves time. <i> is the time rate change of Q - a derivative (rate of change, changing rate) of Q with time. When Q does not change, i = 0. Then ItV = 0. Something wrong again.

 We must consider Field. Electric Filed: E (unit: V/m)

QE = QV/m where <m> is distance. The distance from one place to another place. E means the voltage difference between this one place to this another place.


       energy (work)
-----------------------------   =  Force
distance (displacement)

Or we can use the following empirical formulae.

a) Capacitor

Simplified structure


Dielectric is placed between two conducting plates, each of area A and with a separation of d
C = \frac{\varepsilon A}{d}
ε  =  C (in Farad) x d / A. 

C (in F) / meter = I t (=Q) / V x meter


b) Inductor (coil)

Most simplified structure

Cylindrical air-core coil

 L = \frac{1}{l} \mu_0 K N^2 A
  • L = inductance in henries (H)
  • μ0 = permeability of free space = 4\pi × 10−7 H/m
  • K = Nagaoka coefficient[14]
  • N = number of turns
  • A = area of cross-section of the coil in square metrer (m2)
  • l = length of coil in metres (m)
μ0   = L (in Henry) x l (m) / K (just number with no unit) x N2 x A (m2)

L (in H) /meter = V t /I x meter

To put these in   c = 1/√ε0μ0

c2  = 1 / μ0 ε0 = 1/Vt x It / IVx m2  = meter2 / time2

then

c = meter / time (which is velocity)

This reminds me of another equation in which ε and μ appear - LC Resonant frequency.





\omega_0 = {1 \over \sqrt{LC}}
You can check the units and get number / time, which is frequency. Unit checking is one way to be sure of the correctness of the equation and help you to make some discovery.

We have not reached the end. Looks like a waste of time. So jumping to a conclusion without understanding to save time.

-----

From wiki (Maxwell's Equations)

In a region with no charges (ρ = 0) and no currents (J = 0), such as in a vacuum, Maxwell's equations reduce to:
\begin{align} \nabla \cdot \mathbf{E} &= 0 \quad &\nabla \times \mathbf{E} = \ -&\frac{\partial\mathbf B}{\partial t}, \\ \nabla \cdot \mathbf{B} &= 0 \quad &\nabla \times \mathbf{B} = \frac{1}{c^2} &\frac{\partial \mathbf E}{\partial t}. \end{align}
Taking the curl (∇×) of the curl equations, and using the curl of the curl identity ∇×(∇×X) = ∇(∇·X) − ∇2X we obtain the wave equations
\frac{1}{c^2}\frac{\partial^2 \mathbf E}{\partial t^2} - \nabla^2 \mathbf E = 0\,, \quad \frac{1}{c^2}\frac{\partial^2 \mathbf B}{\partial t^2} - \nabla^2 \mathbf B = 0\,,
which identify
c = \frac{1}{\sqrt{ \mu_0 \varepsilon_0}} = 2.99792458 \times 10^8 \, \mathrm{m~s}^{-1}
with the speed of light in free space. In materials with relative permittivity εr and relative permeability μr, the phase velocity of light becomes
v_\text{p} = \frac{1}{\sqrt{ \mu_0\mu_\text{r} \varepsilon_0\varepsilon_\text{r} }}
which is usually less than c.
In addition, E and B are mutually perpendicular to each other and the direction of wave propagation, and are in phase with each other. A sinusoidal plane wave is one special solution of these equations. Maxwell's equations explain how these waves can physically propagate through space. The changing magnetic field creates a changing electric field through Faraday's law. In turn, that electric field creates a changing magnetic field through Maxwell's addition to Ampère's law. This perpetual cycle allows these waves, now known as electromagnetic radiation, to move through space at velocity c.

------

The point of the above explanation is ＜the curl of the curl identity ∇×(∇×X) = ∇(∇·X) − ∇2X we obtain the wave equations.＞.

To understand this you are required to have some knowledge of and are familiarity with Vector Calculus. However, you can get some idea about this without Vector Calculus.

We have repeatedly used the following basic four equations


            dΦ              di
v(t)  = ------  =  L -------   (LI = Φ)
             dt               dt

             dQ              dv
i(t)  =  ------  =  C -------  (CV = Q)
             dt                dt


            dΦ              di                    di
v(t)  = ------  =  L -------  =  mμ0   -----
             dt               dt                    dt


Now we use


               dv
i(t)  = C -----
               dt


             di                      d2v　　　　　　　　　d2v
=  mμ0  -----   = mμ0   C  -----  =  mμ0  mε0 　------
             dt                      dt2                              dt2

                  d2v
=  m2μ0 ε0 　------
                  dt2

This is a kind of the above curl of curl operation though we do not use 'curl' ＜place, space, or field＞ but only time derivatives.

Then

v(t)               d2v
----  =  μ0 ε0 　------
 m2                    dt2

 
And then

1 / μ0 ε0 = 1/Vt x It / IVx m2  = meter2 / time2

μ0 ε0  = meter2 / time2  = velocity 2   

v(t)               1            d2v
----  =    ----------- 　--------
 m2        velocity 2        dt2
 

        1          d2v         v(t)  
------------ -------  -   ------  = 0
velocity 2      dt2           m2   


If we change ＜v＞ to the vector field＜E(v /m)＞


        1          d2E         E(t)  
------------ -------  -   ------  = 0
velocity 2      dt2           m2   
 
 
 This is the same structure of the first part of the above wave equations.


\frac{1}{c^2}\frac{\partial^2 \mathbf E}{\partial t^2} - \nabla^2 \mathbf E = 0\,, \quad \frac{1}{c^2}\frac{\partial^2 \mathbf B}{\partial t^2} - \nabla^2 \mathbf B = 0\,,


Similarly we can get the 2nd part when we take a time derivative twice of i(t).


                 dv                         
i(t)  =  C -------
                  dt

                         d2i
i (t) =  m2μ0 ε0 　------
                          dt2


      1            d2i           i(t)  
------------ -------  -   ------  = 0
velocity 2      dt2           m2   
 

If we change ＜i＞ to the vector field＜H (i /m)＞


Please note B =  μ0H


sptt
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Thursday, March 19, 2015

Sphere and Inverse-squre law

Surface area

From: wiki

The surface area of a sphere is:
A = 4\pi r^2.
Archimedes first derived this formula. (and)

it equals the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r.

I was once thinking of the surface area ( A = 4\pi r^2. ) about one week how to get this (when I did not know that Archimedes had already done it though I thought someone had done it). Why 4 times the area of a circle, which is
\mathrm{Area} = \pi r^2.\,
 Meanwhile the area is obtained by integration of the circumference of a circle, which is

C = 2\pi r = \pi d.\, (where r = radius and d = diameter).

After about one week thinking I found

I must integrate the circumference of a circle four (4) times, not two (2) times from 0 to r as I must see the sphere both horizontally (from the side) and vertically (from the top or from the bottom) to fully integrate it (the circumference of a circle) from 0 to r. This was a big discovery for me. One week was not a waste of time at all.

-----
A = 4\pi r^2.

This formula reminds us of Coulomb's law and Coulomb's constant which has 4π.

From mostly wiki.

Coulomb's law 

Coulomb's law states that:
The magnitude of the electrostatic force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distance between them.[12]
The force is along the straight line joining them. If the two charges have the same sign, the electrostatic force between them is repulsive; if they have different sign, the force between them is attractive.

( Underlined by sptt. And this law, as any other physics laws, is quite universal and found everywhere if you try to find. (*) Mathematically)

A graphical representation of Coulomb's law

Coulomb's law can also be stated as a simple mathematical expression. The scalar and vector forms of the mathematical equation are
|\mathbf F|=k_e{|q_1q_2|\over r^2}\qquad and \qquad\mathbf F_1=k_e\frac{q_1q_2}{{|\mathbf r_{21}|}^2} \mathbf{\hat{r}}_{21},\qquad respectively,

Coulomb's constant

\begin{align} k_e &= \frac{1}{4\pi\varepsilon_0}=\frac{c_0^2\mu_0}{4\pi}=c_0^2\times 10^{-7}\ \mathrm{H\cdot m}^{-1}\\ &= 8.987\,551\,787\,368\,176\,4\times 10^9\ \mathrm{N\cdot m^2\cdot C}^{-2} \end{align}


This exact value of Coulomb's constant ke  comes from three of the fundamental, invariant quantities that define free space in the SI system: the speed of light c0 , magnetic permeability μ0 , and electric permittivity ε0 , related by Maxwell as:
\frac{1}{\mu_0\varepsilon_0}=c_0^2.
Because of the way the SI base unit system made the natural units for electromagnetism, the speed of light in vacuum c0  is 299792458 m s−1, the magnetic permeability μ0  of free space is 4π·10−7 H m−1, and the electric permittivity ε0  of free space is 1 ⁄ (μ0 c2
0 ) ≈ 8.85418782×10−12 F m−1,[1] so that[2]
\begin{align} k_\text{e} = \frac{1}{4\pi\varepsilon_0}=\frac{c_0^2\mu_0}{4\pi}&=c_0^2\times 10^{-7}\ \mathrm{H\ m}^{-1}\\ &= 8.987\ 551\ 787\ 368\ 176\ 4\times 10^9\ \mathrm{N\ m^2\ C}^{-2}. \end{align}
-----
magnetic permeability μ0 , and electric permittivity ε0

These two are very important number as they constitute the equation
\frac{1}{\mu_0\varepsilon_0}=c_0^2.
as well as the meanings or concepts.

(magnetic) permeability μ0    - 4π·10−7 H m−1

(electric) permittivity ε0    - 1 ⁄ (μ0 c2
0 ) ≈ 8.85418782×10−12 F m−1

where μ0  =  4π·10−7 H m−1
 (magnetic) permeability μ0   = the degree of magnetization in vacuum

 (electric) permittivity ε0   = the degree of electric displacement in vacuum

The concept of electrical displacement is also important, which is discovered or thought by Maxwell and lead to the theory of electromagnetism (though not so simple like this short statement).
-----
(*)

Biot–Savart law

The Biot–Savart law is used for computing the resultant magnetic field B at position r generated by a steady current I (for example due to a wire): a continual flow of charges which is constant in time and the charge neither accumulates nor depletes at any point. The law is a physical example of a line integral, being evaluated over the path C in which the electric currents flow. The equation in SI units is[3]
\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int_C \frac{I d\mathbf l\times\mathbf{r}}{|\mathbf{r}|^3}
where d\mathbf{l} is a vector whose magnitude is the length of the differential element of the wire, in the direction of conventional current, \mathbf{r} is the full displacement vector from the wire element to the point at which the field is being computed, and μ0 is the magnetic constant. Alternatively:
\mathbf{B}(\mathbf{r}) = \frac{\mu_0}{4\pi}\int_C \frac{I d\mathbf l\times\mathbf{\hat r}}{|\mathbf{r}|^2}
where \mathbf{\hat{r}} is the unit vector of \mathbf{r}. The symbols in boldface denote vector quantities.
The integral is usually around a closed curve, since electric currents can only flow around closed paths. An infinitely long wire (as used in the definition of the SI unit of electric current - the Ampere) is a counter-example.
To apply the equation, the point in space where the magnetic field is to be calculated is arbitrarily chosen. Holding that point fixed, the line integral over the path of the electric currents is calculated to find the total magnetic field at that point. The application of this law implicitly relies on the superposition principle for magnetic fields, i.e. the fact that the magnetic field is a vector sum of the field created by each infinitesimal section of the wire individually.[4]


sptt




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