Natural Science Forum / Physics / Particle Physics / October 2004

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\author{\copyright\ Copyright 2004 David E. Rutherford \\
All Rights Reserved \\ \\
E-mail: drutherford@softcom.net \\
http://www.softcom.net/users/der555/biotcomp.pdf}

\title{Biot-Savart's Companion}

\date{October 11, 2004}

\begin{document}
\maketitle

The Biot-Savart law states that the element of magnetic field
$d\B$ produced by a short segment $d{\el}$ of wire of arbitrary
shape carrying a steady line current $I$, in SI units, is
\begin{equation}\label{dB}
d{\B} = {\mu_0 \over 4 \pi}\,{I d{\el} \times \rhat \over r^2}
\end{equation}
with magnitude
\begin{equation}\label{magdB}

\end{equation}
where $\theta$ is the angle between d{\el} and $\rhat$.

I believe the Biot-Savart law has a companion law that, to my
knowledge, has gone undiscovered until now. My companion laws to
(\ref{dB}) and (\ref{magdB}) introduce a `scalar' field $H$ whose
element $dH$ has similar form to (\ref{dB}) and
(\ref{magdB}).\footnote{See
http://www.softcom.net/users/der555/newtransform.pdf. The field
$H$ is actually part of my generalized electric field (however, it
is not referred to as $H$, there). Note that, here, I am using a
three-dimensional, non-relativistic treatment, as opposed to the
four-dimensional, relativistic treatment used at the above link.}

The companion law is derived from $\nabla \cdot {\A}$, where
${\A}$ is the vector potential due to a `point' charge $q'$ moving
with constant velocity ${\vv}'$.\footnote{$\nabla \cdot {\A}$ is
physical and, in general, nonzero in my theory.} We first note
that ${\A} = {\vv}'\phi/c^2$, where $\phi = q'/(4 \pi \epsilon_0
r)$ is the scalar electric potential, due to $q'$, at a field
point $P$ a distance $r$ from $q'$.\footnote{It is important to
note that the scalar potential $\phi$, here, is the
three-dimensional part of my four-dimensional potential at
http://www.softcom.net/users/der555/newtransform.pdf, since this
is a three-dimensional treatment.} Thus, after some substitution
we get
\begin{equation}\label{divA}
\nabla \cdot {\A} = \nabla \cdot \left({{\vv}'\phi \over
c^2}\right) = -\,{q' \over 4 \pi \epsilon_0 c^2}\,{{\vv}' \cdot
\rhat \over r^2}
\end{equation}
where $\rhat$ is a unit vector pointing from $q'$ to $P$.

If we refer to $\nabla \cdot {\A}$ here as $H$, and note that
$\epsilon_0\mu_0 = 1/c^2$, we can write (\ref{divA}) as
\begin{equation}\label{H1}
H = -\,{\mu_0 q' \over 4\pi}\, {{\vv}'\! \cdot \rhat \over r^2}
\end{equation}

Since ${\vv}'\! \cdot \rhat$ is a scalar quantity (in
three-dimensional space), we could also write (\ref{H1}) as
\begin{equation}\label{H2}
H = -\,{\mu_0 q' \over 4\pi}\, {v'\! \cos\theta \over r^2}
\end{equation}
where $v'$ is the magnitude of ${\vv}'$ and $\theta$ is the angle
between ${\vv}'$ and $\rhat$.

The element $dH$ at $P$ due to a `point' charge $dq'$ is
\begin{equation}\label{dH1}
dH = -\,{\mu_0 dq' \over 4 \pi}\, {{\vv}' \cdot \rhat \over r^2}
\end{equation}
or
\begin{equation}\label{dH2}
dH = -\,{\mu_0 dq' \over 4 \pi}\, {v'\! \cos\theta \over r^2}
\end{equation}

Now consider a small segment $d{\el}$ of wire carrying a steady
current $I$, within which the `point' charge $dq'$ is moving with
velocity $\vv'$ parallel to $d{\el}$. In terms of the current $I$,
substituting $dq' = I dt$ and ${\vv}' = d{\el}/dt$, (\ref{dH1})
and (\ref{dH2}) become
\begin{equation}\label{dH3}
dH = -\,{\mu_0 \over 4 \pi}\, {I d{\el} \cdot \rhat \over r^2}
\end{equation}
or
\begin{equation}\label{magdH3}
dH = -\,{\mu_0 \over 4 \pi}\, {I dl \cos\theta \over r^2}
\end{equation}
where $dl$ is the magnitude of $d{\el}$. The equations (\ref{dH3})
and (\ref{magdH3}) are my companions to (\ref{dB}) and
(\ref{magdB}), respectively.

The \emph{additional force} ${\F}_a$ on a test charge $q$ at point
$P$ moving with velocity $\vv$ in the field $H$ at $P$
is\footnote{See http://www.softcom.net/users/der555/actreact.pdf.}
\begin{equation}\label{F}
{\F}_a = -\,q{\vv}H
\end{equation}
If $H$ at $P$ is due to a `point' charge $q'$ moving with constant
velocity ${\vv}'$, we can substitute (\ref{H1}) or (\ref{H2}) into
(\ref{F}), to obtain
\begin{equation}
{\F}_a = -\,q {\vv} \left(-\,{\mu_0 q' \over 4\pi}\, {{\vv}'\!
\cdot \rhat \over r^2}\right) = {\mu_0 q q' \over 4 \pi}\, {{\vv}
\left({\vv}'\! \cdot \rhat \right) \over r^2}
\end{equation}
or
\begin{equation}
{\F}_a = {\mu_0 q q' \over 4 \pi}\, {{\vv} \left(v'\!
\cos\theta\right) \over r^2}
\end{equation}
respectively.

For a steady line current $I$, the element of force $d{\F}_a$ on
$q$ due to a short segment $d{\el}$ of wire, using (\ref{dH3}), is
\begin{equation}\label{dF}
d{\F}_a = -\,q {\vv} dH = {\mu_0 q {\vv} \over 4 \pi}\, {I d{\el}
\cdot \rhat \over r^2}
\end{equation}
The total force ${\F}_a$ on $q$ can be found by integrating
(\ref{dF}) along the wire, resulting in
\begin{equation}
{\F}_a = {\mu_0 q I {\vv} \over 4 \pi} \int{{d{\el} \cdot \rhat
\over r^2}}
\end{equation}

It is interesting to note that the magnetic force ${\F}_m = q
{\vv} \times {\B}$ changes the \emph{direction} of the velocity of
a test particle, but not the magnitude of its velocity. However,
in contrast, my additional force ${\F}_a = -\,q {\vv} H$ changes
the \emph{magnitude} of the velocity, but not its direction. These
two forces go hand-in-hand, along with the electric force ${\F}_e
= q {\E}$, to complete the three-dimensional, non-relativistic
equations of motion of a test particle.\footnote{See
http://www.softcom.net/users/der555/actreact.pdf.}

My equations predict new physics, however, to my knowledge, there
have been no tests conducted that would verify or falsify my laws,
to date, and I have devised none, at this time.

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