SOLUTION OF POISSON'S EQUATION IN INSULATED SPHERE

TO FIND THE ELECTRIC POTENTIAL INSIDE THE CHARGED INSULATING SPHERE

For this purpose most suitable method is solving Poisson’s equations, because inside the insulating charged sphere (ICS) there is a constant charge density and it is also an easy method.

The Poisson’s equation, ∆ ²V= -  Ω/€₀  Where, Ω is the volume charge density. In ICS potential is the function of polar distance(r). So we can change Laplacian of V, using vector differential rule.

∆²V= d²V/dr² + (2/r)dV/dr = -Ω/€₀      or

(r) d²V/dr² + 2dV/dr = -rΩ/€₀

To solve this differential equation by Integral series.

Let, V = c₀ + c₁r + c₂r² + c₃r³ + c₄r⁴+ c₅r⁵ + …………………………..

Then,

dV/dr = c₁ + 2c₂r + 3c₃r² + 4c₄r³ + 5c₅r⁴ + ………………………………

d²V/dr² = 2c₂ + 6c₃r + 12c₄r³ + 20c₅r⁴ + …………………………………….

Therefore the equation,

(r) d²V/dr² + 2dV/dr = -rΩ/€₀

Will be equals to

[2c₂r + 6c₃r² + 12c₄r³ + 20c₅r⁴ + ………..] + [2c₁ + 4c₂r + 6c₃r² + 8c₄r³ + 10c₅r⁴ + …………….]

To equating the coefficient of r, we get c₂ = -Ω/6€₀

And also equating the coefficients of     constant, r, r²,r³,r⁴ to zero

We get,

      c₁ = c₃ = c₄ = 0    and so on.

Therefore the required solution

V = c0 - r 2Ω/6€0

In ICS,

  Ω =  3Q/(4πr³)

At the surface of the sphere, V is V₀ = (Q /[4πR€₀]) and r = R;

V₀ = c₀ – R²Ω/6€₀ = c₀ – QR² /[8πR³€₀]

ie, (Q /[4πR€₀]) = c₀ – QR² /[8πR³€₀]    and c₀ = 3Q /[8πR€₀]

Therefore the potential inside the insulated sphere is equals to,

V = Q /[8πR€0] {3 – (r2/R2)}

It derived by myself therefore i don't sure that this is the solution, so kindly forgive to me.  

MAGNETIC FIELDS IN MAGNETS

I said that I likes the theoretical physics, because it only gives the answers of my doubts. Here i show that how can explain the magnetic field produced by the magnet. But, it has no scientific basis it is only my argument.
Before my presentation i give some recalls:

1.     Magnetic fields are represented by close loops, because to show that they have no begins and end, in magnets they oriented as N-S-N.
2. The production of magnetic field only responsible for current.
3. Current can produced by relative motion of chareges

Here, I solve my two problems,

• How magnetic field produced by around the magnets.
• How the magnetic strength more at the poles.

Look, you can see beautiful pattern when the iron dust powdered around the magnets.
The curved lines are magnetic filed lines(or force lines).

I can show how they originates, from my f recall 2 i said that current only responsible for magnets, but where is the current in magnet. Current is the flow of charge(electrons).So every substances have electrons   they moves around orbit and spins its own axis and produce a current(Ampere's  current).Therefore they produced magnetic fields so you can ask then every substance could produce magnetic fields, but they won't. Yes of course your question is correct, there is a problem the all electrons doesn't revolves around the atom at same plane, therefore most of the substance the magnetic fields cancels each other. But some substance there exists small fields, they are the responsible for magnetic fields in some substances.But there is another current and field by its own spin. But it is only possible in substances having odd number of electron because if it is even the all electron will paired and cancel its field. Then i come back to my problem, look at the below figure.

fig 1

It  shows the magnetic field produced by the atom in the magnetic substance.The center is shows a current element and the arrow shows the magnetic field direction. There are several loop in the magnet we can figure out the a few loops in magnet the figure 2 shows that.The square is show the small portion of magnet and you can see a loop of marked as violet arrow shows the effective magnetic loop, how the effective  field is involve, that is simple the loop surrounded by the magnet.
there are several portions like the fig.2 in the magnet which gives high intense magnetic field just like the fig.3 from this figure we can easily noticed that number of magnetic loops are higher at the poles. This is the reason for the magnetic field intensity greater at poles.     

figure 2

In the figure 3 loops are circulated N to S to N.

the loop passing through S to N through inside the magnet.

WHO AM I?


       I am an undergraduate student(I will be graduate on 2012), studying Bsc.Physics. I study BSC.Physics not for a job, it is for my passion. I am more enthusiastic in the world of physics, that is the explanation of many natural phenomenon by physics is more amazing, and it tells about unbelievable facts in nature(I believe that facts because i studying physics). But i am more interested explanation of cosmic mysteries, i always interested all branches of physics but my adventurous mind always seeking for the unsolved fields. I didn't care about the experimental idea, because i know that a saint{"who seeking for the truth, someone says he is an scientist"} doesn't believe which he(" /she") saw, hear and words, only believes the facts which he discovered. Experiment is only for satisfaction. So i love theoretical physics.
           Someone thinks that, what is the importance for studying theoretical physics, there is any profit for this thing. Yes of course, there is no profit, then i will tell you something what is the profit for watching films, reading books, drinking alcohol etc.. Every life in this world struggle live, they only want food so why want this? yes i know most of them for entertainment.Thats all,
theoretical physics for just entertainment( but to me it is  more...).
       I think you are also interested in physics, interested i means not easy just interested. So you also share your interest with me.