Formula: Lorentz Force Between Two Current-Carrying Wires Magnetic force Electric current Length Distance
$$\class{green}{F} ~=~ \frac{\mu_0 \, L}{2 \pi} \, \frac{ \class{blue}{I_1} \, \class{blue}{I_2} }{r}$$
$$\class{green}{F} ~=~ \frac{\mu_0 \, L}{2 \pi} \, \frac{ \class{blue}{I_1} \, \class{blue}{I_2} }{r}$$
$$\class{blue}{I_1} ~=~ \frac{2\pi \, r}{\mu_0 \, L} \, \frac{ \class{green}{F} }{ \class{blue}{I_2} }$$
$$\class{blue}{I_2} ~=~ \frac{2\pi \, r}{\mu_0 \, L} \, \frac{ \class{green}{F} }{ \class{blue}{I_1} }$$
$$L ~=~ \frac{2\pi \, r}{\mu_0} \, \frac{ \class{green}{F} }{ \class{blue}{I_1} \, \class{blue}{I_2} }$$
$$r ~=~ \frac{\mu_0 \, L}{2 \pi} \, \frac{ \class{blue}{I_1} \, \class{blue}{I_2} }{ \class{green}{F} }$$
$$\mu_0 ~=~ \frac{2\pi \, r \, \class{green}{F}}{L \, \class{blue}{I_1} \, \class{blue}{I_2}}$$
Magnetic force
$$ \class{green}{F} $$ Unit $$ \mathrm{N} $$
The magnetic part of the Lorentz force acts on a current-carrying wire because it is in the magnetic field generated by the other current-carrying wire.
Parallel, thin wires experience an equal force. Depending on the direction of the electric current, the force is either repulsive or attractive.
Electric current
$$ \class{blue}{I_1} $$ Unit $$ \mathrm{A} $$
Charge per unit time flowing through the first wire.
Electric current
$$ \class{blue}{I_2} $$ Unit $$ \mathrm{A} $$
Charge per unit time flowing through the second wire.
Length
$$ L $$ Unit $$ \mathrm{m} $$
The longer the wire, the larger is the Lorentz force on this conductor. Here it is assumed that the two wires are equal in length!
Distance
$$ r $$ Unit $$ \mathrm{m} $$
Distance between the two parallel wires.
Vacuum permeability
$$ \mu_0 $$ Unit $$ \frac{\mathrm{Vs}}{\mathrm{Am}} = \frac{ \mathrm{kg} \, \mathrm{m} }{ \mathrm{A}^2 \, \mathrm{s}^2 } $$
The vacuum permeability is a physical constant and has the following experimentally determined value:
$$ \mu_0 ~=~ 1.256 \, 637 \, 062 \, 12 ~\cdot~ 10^{-6} \, \frac{\mathrm{Vs}}{\mathrm{Am}} $$
Number Pi
$$ \pi $$ Unit $$ - $$
Number Pi is a mathematical constant and has the value \( \pi ~=~ 3.1415926... \).