- #1
chemfirus
- 3
- 1
Now, i am trying to prove that electric field inside spherical shell whose constant surface charge density [tex]\rho[/tex] is zero. Usually, it is proved by using Gauss law (there is no charge enclosed inside gaussian surface so there is no electric field inside the sphere). But, whe i am trying to solve schaum electromagnetics problem (actually, it is not my homework and i don't have to do it) no. 2.53 i was asked to prove it. Because in the chapter is not explained gauss law yet, i try to prove it. Next is my step:
1) I would like to prove it in spherical coordinate. So, i need to use next equation:
d2 = R12 + R22 - 2*R1*R2[cos[tex]\theta[/tex]1*cos[tex]\theta[/tex]2 + sin[tex]\theta[/tex]1*sin[tex]\theta[/tex]2*(cos([tex]\phi[/tex]1-[tex]\phi[/tex]2))].....(1)
Equation (1) is used to find distance between two points (or vector), [tex]\vec{R1}[/tex] and [tex]\vec{R2}[/tex], in spherical coordinate. I would use it as R in coulomb's law.
2) The next step is to model the equation in coulomb's law:
([tex]\rho[/tex]/(4*[tex]\pi[/tex]*[tex]\epsilon[/tex]))*[tex]\oint[/tex](R12sin[tex]\theta[/tex]1/(R12 + R22 - 2*R1*R2*cos[tex]\theta1[/tex])) d[tex]\theta[/tex]1d[tex]\phi[/tex]1...(2)
In equation (2), i have made [tex]\theta[/tex]2 = 0 because i put the evaluated point at [tex]\theta[/tex] = 0 (or simply at z axis in cartesian coordinate). R1 is distance between dS and centre point, R2 is distance between evaluated points with centre points. So, distance between dS and evaluated point got by using equation (1).
I have tried to integrate the equation (2) and i can't prove that the result is zero. I got a natural logaritmic equation at first integral (integrate with d[tex]\theta[/tex]1). When i substitute upper and lower limit with [tex]\pi[/tex] and 0 i don't get zero result. I don't try the second integral because i am not sure i would get the result i want.
Suddenly, i aware that i don't use unit vector yet in equation (2). It makes the distance not become (R12 + R22 - 2*R1*R2*cos[tex]\theta1[/tex])(3/2) like equation we usually met. But, i don't know what is the vector in spherical coordinate. How to write it and how to evaluate it? I stop to use spherical coordinate and try to use cartesian coordinate. But, it is extremely difficult even only for evaluating dS . I get a form which i can't use my integral technique. I really...really stuck now. You need to know that two equations above is equations i derive alone. So, ther is possibility to be wrong.
Which is wrong from my steps? Is the equation? Or model? Could you help me?
1) I would like to prove it in spherical coordinate. So, i need to use next equation:
d2 = R12 + R22 - 2*R1*R2[cos[tex]\theta[/tex]1*cos[tex]\theta[/tex]2 + sin[tex]\theta[/tex]1*sin[tex]\theta[/tex]2*(cos([tex]\phi[/tex]1-[tex]\phi[/tex]2))].....(1)
Equation (1) is used to find distance between two points (or vector), [tex]\vec{R1}[/tex] and [tex]\vec{R2}[/tex], in spherical coordinate. I would use it as R in coulomb's law.
2) The next step is to model the equation in coulomb's law:
([tex]\rho[/tex]/(4*[tex]\pi[/tex]*[tex]\epsilon[/tex]))*[tex]\oint[/tex](R12sin[tex]\theta[/tex]1/(R12 + R22 - 2*R1*R2*cos[tex]\theta1[/tex])) d[tex]\theta[/tex]1d[tex]\phi[/tex]1...(2)
In equation (2), i have made [tex]\theta[/tex]2 = 0 because i put the evaluated point at [tex]\theta[/tex] = 0 (or simply at z axis in cartesian coordinate). R1 is distance between dS and centre point, R2 is distance between evaluated points with centre points. So, distance between dS and evaluated point got by using equation (1).
I have tried to integrate the equation (2) and i can't prove that the result is zero. I got a natural logaritmic equation at first integral (integrate with d[tex]\theta[/tex]1). When i substitute upper and lower limit with [tex]\pi[/tex] and 0 i don't get zero result. I don't try the second integral because i am not sure i would get the result i want.
Suddenly, i aware that i don't use unit vector yet in equation (2). It makes the distance not become (R12 + R22 - 2*R1*R2*cos[tex]\theta1[/tex])(3/2) like equation we usually met. But, i don't know what is the vector in spherical coordinate. How to write it and how to evaluate it? I stop to use spherical coordinate and try to use cartesian coordinate. But, it is extremely difficult even only for evaluating dS . I get a form which i can't use my integral technique. I really...really stuck now. You need to know that two equations above is equations i derive alone. So, ther is possibility to be wrong.
Which is wrong from my steps? Is the equation? Or model? Could you help me?