Deriving Biot-Savart's Law, kinda a math problem not really physics

[SammyS]

Here's my solution:
View attachment 33843

I didn't do the integration, but my solution agrees with this.

But, I do think there is a problem with both solutions.

It's late, so I'll look at it tomorrow.

Below added in Edit:

The concern I have with my solution, is that the variable, y, was used in two distinct ways. The primary way was as the variable of integration. The other way y was used was as a coordinate for the point, P.

I intend to redo my solution using (x₀, y₀) for the coordinates of P.


.

 

[jhae2.718]

This is going to sound unrelated, but how did you make that pdf that is so compatiable with LaTeX?

I used LaTeX :)

MikTeX 2.9 on Windows.

Source:

\documentclass[10pt,letterpaper]{article}
\usepackage[latin1]{inputenc}
\usepackage{amsmath}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{fullpage}
\usepackage{graphicx}
\usepackage{float}
\usepackage{xfrac}
\newcommand{\custpic}[4]{\begin{figure}[H]
	\begin{center}
		\includegraphics[width=#2\textwidth]{#1}
		\vspace{-15pt}
		\caption{#3}
		\vspace{-20pt}
		\label{#4}
	\end{center}
\end{figure}}

\begin{document}
\custpic{pf5}{.75}{Problem diagram}{pf}

Start with Eq. (\ref{bs}), the Biot-Savart Law:
\begin{equation}\label{bs}
	d\vec{B} = \frac{\mu_0}{4\pi}\frac{id\vec{s} \times \hat{r}}{r^2}
\end{equation} 
From the diagram:
\begin{align*}
	r = \sqrt{x^2 + y^2} \\
	x = r\sin(\theta) \\
	y = r\cos(\theta) \\
	d\vec{s} = d\vec{y} 
\end{align*}
Substitute into Eq. (\ref{bs}):
\begin{equation}
	d\vec{B} = \frac{\mu_0}{4\pi}\frac{id\vec{y} \times \hat{r}}{x^2 + y^2}
\end{equation}
Taking the cross product, $id\vec{y} \times \hat{r} = dy\sin(\theta) (-\hat{k})$. Substitute in:
\begin{equation}
	d\vec{B} = \frac{\mu_0 i}{4\pi}\frac{dy\sin(\theta)}{x^2 + y^2} (-\hat{k})
\end{equation} 
From above, $\sin(\theta) = \dfrac{x}{r}$:
\begin{equation}
	d\vec{B} = \frac{\mu_0 i}{4\pi}\frac{xdy}{\left(x^2 + y^2\right)^{\sfrac{3}{2}}} (-\hat{k})
\end{equation} 
Find $\vec{B}$ by taking the integral over the infinite line:
\begin{equation}
	\vec{B} = \frac{\mu_0 i}{4\pi}\int\limits_{-\infty}^{\infty}\frac{xdy}{\left(x^2 + y^2\right)^{\sfrac{3}{2}}} (-\hat{k})
\end{equation} 
\begin{equation}
	\vec{B} = 2 \frac{\mu_0 i}{4\pi}\int\limits_{0}^{\infty}\frac{xdy}{\left(x^2 + y^2\right)^{\sfrac{3}{2}}} (-\hat{k})
\end{equation} 
Integrate (I used Wolfram$|$Alpha) to get:
\begin{equation}
	\vec{B} = 2 \frac{\mu_0 i}{4\pi}\left[\frac{y}{x\sqrt{x^2 + y^2}}\right]_{0}^{\infty} (-\hat{k})
\end{equation} 
The indefinite integral evaluates to:
\begin{equation}
	\vec{B} = \frac{\mu_0 i}{2\pi x} (-\hat{k})
\end{equation}
We can confirm this using Ampere's Law (left as an exercise to the reader...)
\end{document}

 

[flyingpig]

It's getting late here, I will recheck my math tomorrow, thank you to both of you very much. That sentence did not make any sense, but hopefully you understood my appreciation.

 

[flyingpig]

How did you save the file so perfectly? Whenever I saved it as a pdf, and I try reopen it it says it can't read it.

 

[SammyS]

...
The concern I have with my solution, is that the variable, y, was used in two distinct ways. The primary way was as the variable of integration. The other way y was used was as a coordinate for the point, P.
...

Simply use x₀ for x, and use (y₀ - y) for y.

$$\vec{B}(x_0,\,y_0)=-\,\frac{x_0\mu_0 I}{4\pi}\,\hat{k}\, \int \frac{ dy }{(x_0^2 + (y_0-y)^2)^{3/2}}$$

I think you may find it handy to evaluate the integral from y₀ to y .

 

[jhae2.718]

How did you save the file so perfectly? Whenever I saved it as a pdf, and I try reopen it it says it can't read it.

You need to run latex on it.

Simply use x₀ for x, and use (y₀ - y) for y.

$$\vec{B}(x_0,\,y_0)=-\,\frac{x_0\mu_0 I}{4\pi}\,\hat{k}\, \int \frac{ dy }{(x_0^2 + (y_0-y)^2)^{3/2}}$$

I think you may find it handy to evaluate the integral from y₀ to y .

That is an excellent point, and is something I just glossed over.

 

[flyingpig]

WHat dose that mean "run latex" over it...?

 

[jhae2.718]

If you have a version of LaTeX installed, open up the command prompt, change directory to the location where you have saved the file, and type:

pdflatex filename.tex

replacing filename with the actual filename.

 

[flyingpig]

I am guessing LaTeX is not the same thing as MikTeX lol

 

[jhae2.718]

MikTeX is a LaTeX distribution. http://docs.miktex.org/faq/faq.html

Try a LaTeX editor like TeXniCenter, Texmaker, etc to actually write/compile the code if you don't like the CLI.