OUJ Physics Special Course “Time Space and Place Physics” p.15, 16

(Tamiaki YONEYA, Junichiro KISHINE) The force among the Straight Currents are, \begin{equation} f = \frac{\mu_0}{2\pi}\frac{II_1}{l} \end{equation} \begin{equation} (I: Test Current, I_1: Current, l: Current Length, \frac{\mu_0}{2\pi}: Decided by the Unit of Current) \end{equation}
If we put the Movable Electron’s average vector v, \begin{equation} I = Ne|\boldsymbol{v}| \end{equation} (N: Numbers of the Conduction Electron per 1 unit of Wire, e: Charge of Electron)

∴ by (1), (2) \begin{equation} f = \frac{\mu_0}{2\pi}\frac{Ne|{v}|I_1}{l} \end{equation} \begin{equation} F = \frac{f}{N} = \frac{\mu_0}{2\pi}\frac{e|{v}|I_1}{l} \end{equation} With (2) \begin{equation} |v| =\frac{I}{Ne} \end{equation} If we put it in (3) \begin{equation} \frac{f}{N} = \frac{\mu_0}{2\pi}\frac{I}{N}\frac{I_1}{l} \end{equation} \begin{equation} \frac{I}{N} = \frac{I}{Ne}|q| = |q||v| \end{equation} \begin{equation} ∵ |q| = e, |\boldsymbol{v}| = \frac{I}{Ne} \end{equation} Therefore, by (4) \begin{equation} F = \frac{f}{N} = \frac{\mu_0}{2\pi}\frac{I}{N}\frac{I_1}{l} = \frac{\mu}{2\pi}|q||v|\frac{I_1}{l} \end{equation} \begin{equation} F = q\boldsymbol{v}\boldsymbol{B} より \boldsymbol{B} = \frac{F}{q\boldsymbol{v}} \end{equation} from (8), (9), \begin{equation} |\boldsymbol{B}| = \frac{\mu_0}{2\pi}\frac{I_1}{l} \end{equation}