Potential of a rotationally symmetric charge distribution

  • #1
deuteron
57
13
Homework Statement
find the potential caused by the rotationally symmetric charge distribution ##\rho_{\vec r_q}\equiv \rho_{r_q}##
Relevant Equations
##\phi_{e;\vec r} = \iiint\limits_{\R^3} d^3r\ \rho_{\vec r_q} \frac 1 {|\vec r-\vec r_q|}
First, we rewrite the term ##|\vec r-\vec r_q|## in the following way:

$$|\vec r-\vec r_q|= \sqrt{(\vec r-\vec r_q)^2} = \sqrt{\vec r^2 + \vec r_q^2 -2\vec r\cdot\vec r_q} = \sqrt{r^2 + r_q^2 -2rr_q\cos\theta}$$

Due to rotational symmetry, we go to spherical coordinates:

$$\phi_{e;\vec r_q} = \int\limits_0^{2\pi}d\varphi \int\limits_0^\pi d\theta \int\limits_0^\infty dr_q\ r_q^2 \sin^2\theta\ \rho_{r_q} \frac 1 {\sqrt{ r^2 + r_q^2 - 2rr_q\cos\theta}}= 2\pi \int\limits_0^\pi d\theta \int\limits_0^\infty dr_q\ r_q^2 \sin^2\theta\ \rho_{r_q} \frac 1 {\sqrt{ r^2 + r_q^2 - 2rr_q\cos\theta}}$$

For the ##\theta## integral, we do the substitution ##\theta\mapsto\cos\theta,\ d\theta \mapsto (-\frac 1 {\sin\theta})d\cos\theta##, and we get:

$$= 2\pi \int\limits_{1}^{-1} d\cos\theta \int\limits_0^\infty dr_q \ (-\frac 1 {\sin\theta})\ r_q^2 \sin^2\theta\ \rho_{\vec r_q} \frac 1{\sqrt{r^2 + r_q^2 -2rr_q\cos\theta}}\\ = 2\pi \int\limits_{-1}^1 d\cos\theta \int\limits_0^\infty dr_q\ r_q^2 \sin\theta \ \rho_{\vec r_q} \sqrt{r^2 + r_q^2 -2rr_q\cos\theta}^{-1} \\ = 2\pi \int\limits_0^\infty dr_q\ r_q^2 \rho_{\vec r_q} [\frac 1 {rr_q} (r^2 + r_q^2 -2rr_q\cos\theta)^{\frac 12}]_{\cos\theta=-1}^{\cos\theta=1},$$

which leads us to the right answer, at least this is what I have in my solution sheet.

What I don't understand is, when we do the ##\theta##-substitution, why don't we write for the ##\cos\theta## inside the square root as ##\cos(\cos\theta)##, since I would expect the substitution to affect the ##\theta## inside another function too.
 
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  • #2
The substitution is ##u = \cos\theta##, implying that ##du = d\cos\theta = -\sin\theta\, d\theta##. It is usually much clearer to introduce a simple new variable rather than using a function of the original variable to represemt the new one. ##\theta \to \cos\theta## does not mean you replace ##\theta## by ##\cos \theta##, it means you use ##\cos\theta## as your integration variable instead of ##\theta##.

The 3D volume element also contains only a single ##\sin\theta##.
 
  • #3
Orodruin said:
The substitution is ##u = \cos\theta##, implying that ##du = d\cos\theta = -\sin\theta\, d\theta##. It is usually much clearer to introduce a simple new variable rather than using a function of the original variable to represemt the new one. ##\theta \to \cos\theta## does not mean you replace ##\theta## by ##\cos \theta##, it means you use ##\cos\theta## as your integration variable instead of ##\theta##.

The 3D volume element also contains only a single ##\sin\theta##.
Thank you! But I am still confused about why we are not changing the ##\theta## that is already inside the ##\cos##. For example, a substitution like ##x\mapsto \sin \theta ## would look like the following:

$$ \int \sqrt{1-x^2}\ dx \stackrel{x\mapsto \sin \theta} {=} \int \sqrt{1-\sin ^2\theta } \cos\theta\ d\theta = \int\cos^2\theta\ d\theta$$

where we substitute in the function ##f(x)=\sqrt{1-x^2}##

however, in the above example, if we let ##f(x)=r_q^2\ \sin\ \frac1 {\sqrt{r^2 + r_q^2 + \cos\ }}##, I would expect the substitution ##\theta\mapsto \cos\phi## to affect the ##\theta## in the ##\sin## and ##\cos## too, which would look something like:

$$\int\limits_0^\pi d\theta \ r_q^2 \sin\theta \frac 1 {\sqrt{r^2 + r_q^2 + \cos\theta}}\stackrel{\theta\mapsto \cos\phi}{=} \int\limits_{1}^{-1} d\cos\phi \ (-\frac 1{\sin\phi}) r_q^2 \sin(\sin\phi) \frac 1 {\sqrt{r^2 + r_q^2 + \cos(\sin\phi)}}$$

I still haven't understand why we don't change the variables **in** the ##\cos## and ##\sin## when we do a substitution
 
  • #4
deuteron said:
But I am still confused about why we are not changing the θ that is already inside the cos.
You are changing it. If ##u = \cos\theta## then ##\theta = \acos u## so ##\cos\theta = \cos(\acos u) = u##.
 
  • #5
Thank you very much!!
 

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