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.

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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}
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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).
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(*)

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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