Magnetic force on a wire due to a loop

[Ngineer]

Homework Statement

Calculate the net magnetic force on I2 due to I1

Relevant Equations

Fmag = I * ([a] x [B1])
where:
[a] is a vector representing the length and direction of the wire carrying I2.
[B1] is the magnetic field vector due to I1 (Please see figure.)

I just need to confirm my intuition that the magnetic force on the wire carrying I2 is 0.
Basis for my intuition:

* Right above the center of the loop carrying I1, the magnetic field lines are in exactly the same direction as the piece of wire carrying I2, so [a] x [B1] = 0.

(photo from HyperPhysics)

* I2's own magnetic field is curling around it (Biot-Savart law and Ampere's law), and never goes through it, meaning an external field that actually goes through the wire will have no effect on it.

Note: only the shown section of the wire carrying current I2 is to be considered in calculations.

Is my intuition correct?
Thank you

 

[kuruman]

Your reasoning correctly supports your intuition.