Problem n. 7 chapter 4 Eisberg Resnick "Quantum Physics"

[baffetto59]

Homework Statement

the solution is not the same stated by the authors. Asking help about

Relevant Equations

integral

a

 

[renormalize]

Homework Statement: the solution is not the same stated by the authors. Asking help about
Relevant Equations: integral

Please state the problem exactly as written and post it here in LaTeX (see the guide below) so readers don't have to open an attachment. To receive help, you also need to display your attempted solution.

 

[baffetto59]

\documentclass{article}

\title {Problem n. 7 chapter 4 Eisberg Resnick "Quantum Physics"}
\usepackage{graphicx}
\graphicspath{ {./images/} }
\begin{document}
\maketitle
\begin{description}
\item Show that the number of $\alpha$ particles scattered by an angle $\Theta$ or greater in Rutherford scattering is
\item $(\frac{1}{4\pi\epsilon_0})^2\pi I\rho t(\frac{zZe^2}{Mv^2})^2\cot^2(\Theta/2)$\space(1)
\item SOLUTION
\item starting from Rutherford formula dN=$(\frac{1}{4\pi\epsilon_0})^2\pi I\rho t(\frac{zZe^2}{Mv^2})^2\frac{1}{\sin^4(\Theta/2)}d\Omega$ \space(1)
\item $d\Omega=2\pi\sin(\Theta)d\Theta$
\item integrate (1) from $\Theta$ to $\pi$
\item N=$(\frac{1}{4\pi\epsilon_0})^2\pi I\rho t(\frac{zZe^2}{Mv^2})^2 \int_\Theta^\pi\frac{2\pi\sin(\Theta)d\Theta}{\sin^4(\Theta/2)}$
\item let u=$\Theta/2$\space;\space$d\Theta=2du$\space;\space$\sin(\Theta)=2\sin(\Theta/2)\cos(\Theta/2)$
\item =$(\frac{1}{4\pi\epsilon_0})^2\space 8\pi^2I\rho t(\frac{zZe^2}{Mv^2})^2 \int\frac{\sin(u)\cos(u)}{\sin^4(u)}du$
\item =$(\frac{1}{4\pi\epsilon_0})^2\space 8\pi^2I\rho t(\frac{zZe^2}{Mv^2})^2 \int\frac{\cos(u)}{\sin^3(u)}du$
\item =$(\frac{1}{4\pi\epsilon_0})^2\space 8\pi^2I\rho t(\frac{zZe^2}{Mv^2})^2 \int\frac{d\sin(u)}{\sin^3(u)}$
\item =$(\frac{1}{4\pi\epsilon_0})^2\space 8\pi^2I\rho t(\frac{zZe^2}{Mv^2})^2 (-\frac{1}{2})\frac{1}{\sin^2(\Theta/2)}\mid^\pi_\Theta$
\item =$(\frac{1}{4\pi\epsilon_0})^2\space 8\pi^2I\rho t(\frac{zZe^2}{Mv^2})^2 (-\frac{1}{2})(1-\frac{1}{\sin^2(\Theta/2)})$
\item =$(\frac{1}{4\pi\epsilon_0})^2\space 8\pi^2I\rho t(\frac{zZe^2}{Mv^2})^2 (\frac{1}{2})(\frac{1-sin^2(\Theta/2}{\sin^2(\Theta/2)})$
\item =$\frac{1}{4\epsilon_0^2}\space I \rho t(\frac{zZe^2}{Mv^2})^2\cot^2(\Theta/2)$

\end{description}
\includegraphics{scattering}
\end{document}

 

[BvU]

Yeah, well, that's not what @renormalize meant

But when I compare your

starting from Rutherford formula $$dN=\left (\frac{1}{4\pi\epsilon_0}\right )^2\;\pi I\rho t \left (\frac{zZe^2}{Mv^2}\right )^2\;\frac{1}{\sin^4(\Theta/2)}d\Omega$$

with the book:$$
N(\Theta)\, d\Theta =\left(\frac{1}{4\pi\epsilon_0}\right )^2\;\left(\frac{zZe^2}{2Mv^2}\right )^2\;\frac{ I\rho t\; 2\pi \sin\Theta\ d\Theta}{\sin^4(\Theta/2)} \tag {4-7}$$
I see you have a factor $4\pi$ too many.

$\$

 

[BvU]

Re $\LaTeX$: the button 'LaTex Guide' at the lower left gets you started. In PF you can insert formulas (in-line math and displayed math), but not complete documents.

$\$

 

[BvU]

Problem n. 7 chapter 4 Eisberg Resnick "Quantum Physics"}​

Show that the number of $\alpha$ particles scattered by an angle $\Theta$ or greater in Rutherford scattering is
$$\left (\frac{1}{4\pi\epsilon_0}\right)^{\!2}\pi I\rho t \left (\frac{zZe^2}{Mv^2}\right)^{\!2}\cot^2(\Theta/2)
$$

SOLUTION​

starting from Rutherford formula $$dN=\left (\frac{1}{4\pi\epsilon_0}\right )^{\!2}\left(\frac{zZe^2}{2Mv^2}\right )^{\!2} I\rho t \;\frac{d\Omega}{\sin^4(\theta/2} \tag{4-7}
$$ with $d\Omega=2\pi\sin(\Theta)d\Theta\ . \ \$Integrate $(4{\text -}7)$ from $\Theta$ to $\pi$: $$

N=\left (\frac{1}{4\pi\epsilon_0}\right )^{\!2}\;\left (\frac{zZe^2}{Mv^2}\right )^{\!2}\;\frac{ \pi I\rho t}{2}\; \int_\Theta^\pi\frac{sin(\theta)d\theta}{\sin^4(\theta/2)}
$$ let $u=\theta/2\space;\space d\theta=2du\space;\space\sin(\Theta)=2\sin(\Theta/2)\cos(\Theta/2)\ \ \Rightarrow$ $$
\begin{align*}
\int_\Theta^\pi\frac{sin(\theta)d\theta}{\sin^4(\theta/2)}&=
4\int_{\Theta/2}^{\pi/2}\frac{\sin u\cos u \, du}{\sin^4 u}\\ \ \\ &=\
\left . \frac{-2}{\sin^2(u)}\ \right |_{\Theta/2}^{\pi/2} \ = -2 \left (1-\frac {1}{\sin^2(\Theta/2)}\right )
= 2\cot^2(\Theta/2)
\end{align*}
$$so that $$N = \left (\frac{1}{4\pi\epsilon_0}\right )^{\!2}\;\left (\frac{zZe^2}{Mv^2}\right )^{\!2}\; \pi I\rho t\; \cot^2(\Theta/2)\ .$$as desired,

just practicing my $\TeX$ -- the answer was given away in #4 already

$\$

 

[baffetto59]

Yeah, well, that's not what @renormalize meant

But when I compare your

with the book:$$
N(\Theta)\, d\Theta =\left(\frac{1}{4\pi\epsilon_0}\right )^2\;\left(\frac{zZe^2}{2Mv^2}\right )^2\;\frac{ I\rho t\; 2\pi \sin\Theta\ d\Theta}{\sin^4(\Theta/2)} \tag {4-7}$$
I see you have a factor $4\pi$ too many.

$\$

$d\Omega=2\pi\sin(\Theta)d\Theta$
It's coherent

 

[renormalize]

Please use double-$$ signs for a display equation or double-## signs for an in-line equation, like $$d\Omega=2\pi\sin(\Theta)d\Theta$$:$$d\Omega=2\pi\sin(\Theta)d\Theta$$

 

[BvU]

$d\Omega=2\pi\sin(\Theta)d\Theta$
It's coherent

No. It's wrong.

again: your

starting from Rutherford formula $$dN=(\frac{1}{4\pi\epsilon_0})^2\pi I\rho t(\frac{zZe^2}{Mv^2})^2\frac{1}{\sin^4(\Theta/2)}d\Omega$$

has a $\pi$ too many in the numerator and misses a $2^2$ in the denominator when compared to $(4{\text-}7)$ in the book.

$\$

 

[baffetto59]

Yes, thanks. I started from wrong formula.