Question

PROBLEM4 (a) Write an expression for the mass density ρ(r) of a point particle with a mass m at a position r; How much is the volume integral of ρ over the entire space? (b) What is the mass density of a system of two point particles with mass m each. One of the particle is at the origin of the coordinate system and the other one is at ä. How much is the integral of the density over the entire space? (c) What is the mass density of a uniform infinitesimally small spherical shell of radius R and total mass M, centered at the origin. Show that the integral over the entire space is equal to M.

Answer 1

Mathematically they are written using the Dirac delta function defined as

$\delta(x-a)=\begin{cases} \infty & \text{ if } x= a\\ 0 & \text{ otherwise} \end{cases}$

so that

$\int_{-\infty}^\infty f(x)\delta(x-a)dx=f(a)$

Also, dirac delta function has the dimensions of length inverse. Applying this concept

(a)

$\rho(\vec{r}\,)=m\delta^3(\vec{r}-\vec{r}\,')$

where

$\delta^3(\vec{r}-\vec{r}\,')=\delta(x-x')\delta(y-y')\delta(z-z')$

and

$\vec{r}=\vec{x}+\vec{y}+\vec{z}$.

Volume integral should be m,

$\int\rho(\vec{r}\,)d^3r=\int m\delta^3(\vec{r}-\vec{r}\,')d^3r=m$

(b)Similar to previous case

$\rho(\vec{r}\,)=m\delta^3(\vec{r}\,)+m\delta^3(\vec{r}-\vec{a}\,)$

$\int\rho(\vec{r}\,)d^3r=\int m\delta^3(\vec{r}\,)+m\delta^3(\vec{r}-\vec{a}\,)d^3r=2m$

(c)

$\rho(\vec{r}\,)=\frac{M}{4\pi r^2}\delta(r-R)$

so that

$\int \rho(\vec{r}\,)d^3r=\iiint \frac{M}{4\pi r^2}\delta(r-R)r^2drd\theta d\phi=M$