PX157 - C3a - the biot-savart law

the law

\vec{B} (\vec{r}, t) = \frac{μ_{0}}{4 π} q \frac{\vec{v} (t) \times (\vec{r} - {\vec{r}}^{'} (t))}{| \vec{r} - {\vec{r}}^{'} (t) |^{3}}

where,
$\vec{r} =$ position where $\vec{B}$ is measured
${\vec{r}}^{'} (t) =$ position of the source, charge that is moving
$μ_{0} =$ permeability of vacuum $= 4 π \times 10^{- 7} H m^{- 1}$ ( $H = h e n r y = k g m^{2} c^{- 2}$ )

law is valid for magnetostatics: particle acceleration would give rise to extra terms and radiation

forces from two moving charges

two charges, $q$ , moving along parallel paths separated by a distance, $\vec{r} = r \hat{y}$ , at velocities, ${\vec{v}}_{1}$ and ${\vec{v}}_{2}$ , such that ${\vec{v}}_{1} = {\vec{v}}_{2} = v \hat{x}$
from coulomb's law:
- electrostatic force felt by charge 1 due to charge 2:

{\vec{F}}_{E, 21} = \frac{q^{2}}{4 π ϵ_{0} r^{2}} \hat{y}

- electrostatic force felt by charge 2 due to charge 1:

{\vec{F}}_{E, 12} = - \frac{q^{2}}{4 π ϵ_{0} r^{2}} \hat{y}

from the biot-savart law :
- magnetic field felt by charge 1 due to charge 2:

{\vec{B}}_{2} (a t 1) = \frac{μ_{0}}{4 π} q \frac{{\vec{v}}_{2} \times \vec{r}}{r^{3}} = \frac{μ_{0} q v}{4 π ϵ_{0} r^{2}} \hat{z}

- magnetic field felt by charge 1 due to charge 2:

{\vec{B}}_{1} (a t 2) = \frac{μ_{0}}{4 π} q \frac{{\vec{v}}_{2} \times (- \hat{r})}{r^{3}} = - \frac{μ_{0} q v}{4 π r^{2}} \hat{z}

the lorentz force:
- force felt by charge 1 moving in the magnetic field of charge 2:

{\vec{F}}_{B, 21} = q {\vec{v}}_{1} \times {\vec{B}}_{2} (a t 1) = - \frac{μ_{0} q^{2} v^{2}}{4 π r^{2}} \hat{y}

- force felt by charge 2 moving in the magnetic field of charge 1:

{\vec{F}}_{B, 12} = q {\vec{v}}_{2} \times {\vec{B}}_{1} (a t 2) = \frac{μ_{0} q^{2} v^{2}}{4 π r^{2}} \hat{y}

comparing the magnitudes of the two forces:

\frac{| {\vec{F}}_{B} |}{| {\vec{F}}_{E} |} = μ_{0} ϵ_{0} v^{2}

the speed of light can be defined as:

c = \frac{1}{\sqrt{μ_{0} ϵ_{0}}}

\frac{| {\vec{F}}_{B} |}{| {\vec{F}}_{E} |} = {(\frac{v}{c})}^{2}

for charges with $v << c : {\vec{F}}_{E} >> {\vec{F}}_{B}$