PX157 - C3d - forces between two current loops

two parallel straight wires, each carrying a current in the positive $z$ -direction:
- wire 1 is centred at $x = y = 0$
- wire 2 is centred at $y = a, x = 0$
the magnetic field generated by wire 1:

{\vec{B}}_{1} (\vec{r}) = \frac{μ_{0} I_{1}}{2 π R} \hat{ϕ}

- at wire 2:

{\vec{B}}_{1} (a t 2) = - \frac{μ_{0} I_{1}}{2 π a} \hat{x}

the magnetic field generated by wire 2:

{\vec{B}}_{2} (\vec{r}) = \frac{μ_{0} I_{1}}{2 π | \vec{r} - a \hat{y} |} {\hat{ϕ}}_{2}

- at wire 1:

{\vec{B}}_{2} (a t 1) = \frac{μ_{0} I_{2}}{2 π a} \hat{x}

force per unit length on wire 1 due to ${\vec{B}}_{2} (a t 1) :$

{\vec{F}}_{1} = I_{1} \hat{z} \times {\vec{B}}_{2} (a t 1) = I_{1} \frac{μ_{0} I_{2}}{2 π a} \hat{z} \times \hat{x} = \frac{μ_{0} I_{1} I_{2}}{2 π a} \hat{y}

force per unit length on wire 2 due to ${\vec{B}}_{1} (a t 2) :$

{\vec{F}}_{2} = I_{2} \hat{z} \times {\vec{B}}_{1} (a t 2) = I_{2} \frac{- μ_{0} I_{1}}{2 π a} \hat{z} \times \hat{x} = - \frac{μ_{0} I_{1} I_{2}}{2 π a} \hat{y}

currents in the same direction attract each other and in the opposite directions repel each other