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

[Apashanka]

Homework 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??

 

[Apashanka]

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??

 

[Apashanka]

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...

 

[Apashanka]

Okay I have made a schematic diagram

$\vec r$ 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

 

[PeroK]

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:

http://hyperphysics.phy-astr.gsu.edu/hbase/Mechanics/sphshell.html#wtls

 

[Apashanka]

@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]

@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]

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.

 

[Apashanka]

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

 

[Apashanka]

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

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 ??

 

[Apashanka]

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

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...