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What will be ##\int_{v'}\vec \nabla_{v'}(1/R)dv'##

  • Thread starter Apashanka
  • Start date
387
13
Problem Statement
What will be ##\int_{v'}\vec \nabla_{v'}(1/R)dv##
Relevant Equations
if the source be represented by the vector ##\vec r'## and another point of interest be ##\vec r## where ##\vec R=\vec r-\vec r'## with respect to a fixed origin,
Now from vector theorem ##\nabla_r'(1/R)=\frac{\hat R}{R^2}##
Now what will be ##\int_v'\nabla_r'(1/R)dv'## over a spherical volume....can anyone please help me out??
Problem Statement: What will be ##\int_{v'}\vec \nabla_{v'}(1/R)dv##
Relevant Equations: if the source be represented by the vector ##\vec r'## and another point of interest be ##\vec r## where ##\vec R=\vec r-\vec r'## with respect to a fixed origin,
Now from vector theorem ##\nabla_r'(1/R)=\frac{\hat R}{R^2}##
Now what will be ##\int_v'\nabla_r'(1/R)dv'## over a spherical volume....can anyone please help me out??

if the source be represented by the vector ##\vec r'## and another point of interest be ##\vec r## where ##\vec R=\vec r-\vec r'## with respect to a fixed origin,
Now from vector theorem ##\vec \nabla_{r'}(1/R)=\frac{\hat R}{R^2}##
Now what will be ##\int_{v'}\vec \nabla_{r'}(1/R)dv'## over a spherical volume....can anyone please help me out??
 
387
13
Problem Statement: What will be ##\int_{v'}\vec \nabla_{v'}(1/R)dv##
Relevant Equations: if the source be represented by the vector ##\vec r'## and another point of interest be ##\vec r## where ##\vec R=\vec r-\vec r'## with respect to a fixed origin,
Now from vector theorem ##\nabla_r'(1/R)=\frac{\hat R}{R^2}##
Now what will be ##\int_v'\nabla_r'(1/R)dv'## over a spherical volume....can anyone please help me out??

Problem Statement: What will be ##\int_{v'}\vec \nabla_{v'}(1/R)dv##
Relevant Equations: if the source be represented by the vector ##\vec r'## and another point of interest be ##\vec r## where ##\vec R=\vec r-\vec r'## with respect to a fixed origin,
Now from vector theorem ##\nabla_r'(1/R)=\frac{\hat R}{R^2}##
Now what will be ##\int_v'\nabla_r'(1/R)dv'## over a spherical volume....can anyone please help me out??

if the source be represented by the vector ##\vec r'## and another point of interest be ##\vec r## where ##\vec R=\vec r-\vec r'## with respect to a fixed origin,
Now from vector theorem ##\vec \nabla_{r'}(1/R)=\frac{\hat R}{R^2}##
Now what will be ##\int_{v'}\vec \nabla_{r'}(1/R)dv'## over a spherical volume....can anyone please help me out??
The volume integral is written in terms of surface integral ##\int_s(1/R)\vec ds'##
##R=√[(\vec r-\vec r')•(\vec r-\vec r')]=√(r^2+r'^2-2r•r')=r'√(1+\frac{r^2}{r'^2}-2r•r'/r'^2)## for large ##r' ,R=r'##
##\int_s(1/r')r'^2sin\theta d\theta d\phi\hat r'=4\pi r'##
Is it correct??
 
387
13
The volume integral is written in terms of surface integral ##\int_s(1/R)\vec ds'##
##R=√[(\vec r-\vec r')•(\vec r-\vec r')]=√(r^2+r'^2-2r•r')=r'√(1+\frac{r^2}{r'^2}-2r•r'/r'^2)## for large ##r' ,R=r'##
##\int_s(1/r')r'^2sin\theta d\theta d\phi\hat r'=4\pi r'##
Is it correct??
The volume integral is having an ans. ##\int_{v'}\vec \nabla_{r'}(1/R)dv'=\frac{4\pi}{3}\vec r## over a large spherical volume,where ##\vec R=\vec r-\vec r'## is with respect to the origin coinciding with the centre is the sphere.
Can anyone please help me out in proving this.....
 
387
13
Okay I have made a schematic diagram
IMG_20190512_213054.jpg

##\vec r## is fixed,##\vec R=\vec r-\vec r'##
now ##\int_{v'}\vec \nabla_{r'}(1/R)dv'=\int_{s'}(1/R)\vec ds'## is done over a spherical surface having radius ##r_s## then it reduces to ##\int_{s'}\frac{1}{r_s^2+r^2-2rr_scos(\phi)}r_s^2sin\theta d\theta d\omega\hat r_s=2πr_s\int_{s'}\frac{1}{r_s^2+r^2-2rr_scos(\phi)}sin\theta d\theta(\vec r-\vec R) ## where ##\omega## is the azimuth 0 to 2π ,##\theta=0## to ##\pi##.
Can anyone will give any hints how to deal with such type of integrals??
Note:within the integral ##\vec R=\vec r-\vec r_s## has it's tail at the surface of the sphere
 
Last edited:

PeroK

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Okay I have made a schematic diagram
View attachment 243456
##\vec r## is fixed,##\vec R=\vec r-\vec r'##
now ##\int_{v'}\vec \nabla_{r'}(1/R)dv'=\int_{s'}(1/R)\vec ds'## is done over a spherical surface having radius ##r_s## then it reduces to ##\int_{s'}\frac{1}{r_s^2+r^2-2rr_scos(\phi)}r_s^2sin\theta d\theta d\omega\hat r_s=2πr_s\int_{s'}\frac{1}{r_s^2+r^2-2rr_scos(\phi)}sin\theta d\theta(\vec r-\vec R) ## where ##\omega## is the azimuth 0 to 2π ,##\theta=0## to ##\pi##.
Can anyone will give any hints how to deal with such type of integrals??
Note:within the integral ##\vec R=\vec r-\vec r_s## has it's tail at the surface of the sphere
This is similar to the integration in the shell theorem, which you may have encountered. E.g:

 
387
13
@PeroK
Okk ,then in that case the integral reduces to
##\int_{s'}\frac{1}{r_s^2+r^2-2rr_scos(\theta)}r_s^2sin\theta d\theta d\omega\hat r_s=2πr_s\int_{s'}\frac{1}{r_s^2+r^2-2rr_scos(\theta)}sin\theta d\theta(\vec r-\vec R)=4\pi\vec r-\int_s2\pi r_s\frac{\vec R}{r_s^2+r^2-2rr_scos(\theta)}sin\theta d\theta##
The problem here is within the integral there is ##\hat r_s## which is not constt. throughout the integration
 

PeroK

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@PeroK
Okk ,then in that case the integral reduces to
##\int_{s'}\frac{1}{r_s^2+r^2-2rr_scos(\theta)}r_s^2sin\theta d\theta d\omega\hat r_s=2πr_s\int_{s'}\frac{1}{r_s^2+r^2-2rr_scos(\theta)}sin\theta d\theta(\vec r-\vec R)=4\pi\vec r-\int_s2\pi r_s\frac{\vec R}{r_s^2+r^2-2rr_scos(\theta)}sin\theta d\theta##
The problem here is within the integral there is ##\hat r_s## which is not constt. throughout the integration
The link I gave you shows a technique for tackling that.
 

PeroK

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PS if you have studied electromagnetism then a quick way is to find a charged sphere that generates the same integral for the electric field.
 
387
13
PS if you have studied electromagnetism then a quick way is to find a charged sphere that generates the same integral for the electric field.
I understood this but the problem is how to tackle the unit vector ##\hat r_s## within the integral which is not constt. ,##r_s## is the radius of the sphere of integration
 
387
13
I understood this but the problem is how to tackle the unit vector ##\hat r_s## within the integral which is not constt. ,##r_s## is the radius of the sphere of integration
IMG_20190513_114747.jpg

Our integration is ##\int_s(1/R)\vec ds=\int_s(1/R)r_s^2sin\theta d\theta d\phi \hat r_s,##
##R=√(r^2+r_s^2-2rr_scos\theta)##
##\hat r_s=(\vec r-\vec R)/r_s##
Putting these it reduces to ##2\pi r_s \int_s[\frac{\vec r}{R}-\hat R]sin\theta d\theta##
Now since ##\vec r## is constt. the first part is solvable but how to do the second part ??
 
387
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Okay I am asking for this type of integrals is due to ,I have came across this in the following snap of a paper by @Nick Kaiser
IMG_20190513_205158.jpg

In deriving this equation 8 from eq. 6 I think this integral will come in way for the term in the RHS containing ##\bar n##.


Will anyone please help me out??or any idea in deriving eq.8 from eq.6....
 

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