I Work to move a point charge from infinity to the centre of a charge distribution

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The discussion focuses on calculating the work required to move a point charge from infinity to the center of a charge distribution defined by the electrostatic potential. The potential is expressed using an integral involving the charge density, which is given in terms of constants and an exponential function. There is a debate regarding the correct expression for the distance between the charge and the charge distribution, with some participants questioning the use of spherical coordinates and the factor of 3 in the integral. The consensus is that the factor should be 2, and the distance should be correctly represented to avoid confusion. The conversation concludes with participants clarifying their misunderstandings and expressing gratitude for the guidance provided.
LeoJakob
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Electrostatic potential $$ \Phi(\vec{r})=k \int \mathrm{d}^{3} r \frac{\rho\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} (i) $$ with $$ k=\frac{1}{4\pi\epsilon_{0}} $$ in SI units.
What work is required to move a point charge q from infinity to the center of the through $$ \rho(\vec{r})=\rho_{0}\mathrm{e}^{-a r} (ii) $$ given charge distribution, where $$ a \text{ and } \rho_{0} $$ are constants? Work in Gaussian units.

To solve the problem I would use spherical coordiantes. The potential only depends on the radial difference between the charge q with position
$$\vec{r} = r\hat{e}_r \text{ and the location vector of the charge distribution element } \vec{r'} = r' \hat{e}_r \text{ such that }\\
\left|\vec{r}-\vec{r'}\right| = \sqrt{(r-r')^2}
$$

$$\text{Define } \Phi(\infty)=0 \text{ then the work is given by } W= q \Phi(\vec r)=q \Phi(r)$$

$$W= q\Phi(\vec{r})=k \int \mathrm{d}^{3} r' \frac{\rho\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|}= \int \mathrm{d}^{3} r' \frac{\rho(r^{\prime})}{\sqrt{(r-r')^2}}=4\pi\int \limits_\infty^0 (r')^3 \frac{\rho(r^{\prime})}{\sqrt{(r-r')^2}}dr^{\prime}$$

The factor 4pi comes from the integration in spherical coordinates and k=1 in Gaussian units.

Is my approach right?
 
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I assume the 3 in $$4\pi\int \limits_\infty^0 (r')^3 \frac{\rho(r^{\prime})}{\sqrt{(r-r')^2}}dr^{\prime}$$is a typo ?

I don't agree with
LeoJakob said:
the potential only depends on the radial difference between the charge q with position

And certainly not with ##\left |\vec{r}-\vec{r}^{\prime}\right| = {\sqrt{(r-r')^2}}##

However -- fortunately -- you are only interested in ##\vec r=\vec 0##, so if you fill that in in ##(i)## you're allright.

##\ ##
 
BvU said:
I assume the 3 in $$4\pi\int \limits_\infty^0 (r')^3 \frac{\rho(r^{\prime})}{\sqrt{(r-r')^2}}dr^{\prime}$$is a typo ?

I don't agree withAnd certainly not with ##\left |\vec{r}-\vec{r}^{\prime}\right| = {\sqrt{(r-r')^2}}##

However -- fortunately -- you are only interested in ##\vec r=\vec 0##, so if you fill that in in ##(i)## you're allright.

##\ ##
First of all: Thanks for taking the time to answer me :)

the 3 in $$4\pi\int \limits_\infty^0 (r')^3 \frac{\rho(r^{\prime})}{\sqrt{(r-r')^2}}dr^{\prime}$$is a typo because:

$$W= q\Phi(\vec{r})=k \int \mathrm{d}^{3} r' \frac{\rho\left(\vec{r}^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|}=\int \limits_{0}^{2 \pi} \int \limits_{0}^{\pi} \int \limits_{\infty}^{0} \frac{p\left(r^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|}(\sin \theta) \cdot\left(r^{\prime}\right)^{2} d r^{\prime} d \theta d \phi \\ =\int \limits_{0}^{2 \pi} d \phi \int \limits_{0}^{\pi} \sin \theta d \theta \int \limits_{\infty}^{0}\left(r^{\prime}\right)^{2} \frac{\rho \left(r^{\prime}\right)}{\left|\vec{r}-\vec{r}^{\prime}\right|} d r^{\prime} =4\pi\int \limits_\infty^0 (r')^2 \frac{\rho(r^{\prime})}{\left|\vec{r}-\vec{r}^{\prime}\right|}dr^{\prime} \\ \Rightarrow $$

Would you agree with(?):

$$q\Phi(0)=4\pi\int \limits_\infty^0 (r')^2 \frac{\rho_{0}\mathrm{e}^{-a r^{\prime}})}{\left|\vec{r}^{\prime}\right|}dr^{\prime} $$

Why do you think the following is wrong?

$$\left|\vec{r}-\vec{r}^{\prime}\right|=\left|\left(r-r^{\prime}\right) \vec{e}_{r}\right|=\left|r-r^{\prime}\right| \cdot \underbrace{\| \vec{e}_{r}||}_{=1}=\sqrt{\left(r-r^{\prime}\right)^{2}} $$
 
LeoJakob said:
First of all:

Ah, and I forgot:
:welcome: ##\qquad ##!​

LeoJakob said:
Why do you think the following is wrong?$$\left|\vec{r}-\vec{r}^{\prime}\right|=\left|\left(r-r^{\prime}\right) \vec{e}_{r}\right|=\left|r-r^{\prime}\right| \cdot \underbrace{\| \vec{e}_{r}||}_{=1}=\sqrt{\left(r-r^{\prime}\right)^{2}}$$
Which ##\vec e_r## would that be ? The one from ##\vec r## or the one from ##\vec r'## ?

If ##|\vec r| = |\vec r'|## then ##|\vec r -\vec r'| =0 ## only if ##\theta=\theta'\ \&\ \phi=\phi'## !
1706223420229.png

The ##\hat e_r ## unit vector for ##\vec r -\vec r'## is not along ##\hat e_r## nor is it along ##\hat e_{r'}## !##\ ##
 
Last edited:
Thanks for the welcome :) I now understand my mistake, thank you very much ! :)
 
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