How does one calculate the magnetic force between two coaxial solenoids, placed in a way their currents are in the same sense? There is a simple way to treat both as dipoles and then calculate the force, however since magnetic monopoles are only hypothetical – I wanted to know how it is possible to arrive at this result if we just take the field due to one solenoid – and see the force on the second solenoid due to this. The maths might be very difficult since the field is non-uniform.
The result by using dipoles is: $$F=\frac{\mu}{4 \pi} \frac{6m_1m_2}{r^4}$$ where $m_1$ and $m_2$ are the solenoids' dipole moments, and $r$ is the distance between their centres. This equation is for when the distance between the solenoids is much larger than their radii.
[Physics] Force between two solenoids
electromagnetismforceshomework-and-exercises
Best Answer
The best way is indeed to treat the solenoid as a magnetic dipole, which is a good approximation in the regime where the distance between the solenoids is much larger than the radius. One misconception you seem to have: magnetic monopoles are hypothetical, but magnetic dipoles are very much a real thing, which require no recourse to monopoles to use mathematically.
Anyway, if you want to calculate it exactly, the general method is to take one solenoid, find the field from it far away (which should be a dipole field), and integrate $I\oint d\vec l\times B$ around a loop. In the far field limit, the force on the whole solenoid will just be the force on one loop multiplied by the number of loops in the solenoid (as the magnetic field only depends on the distance away from the dipole and the angle, and if the solenoids are far enough apart the angle does not change appreciably over the length of the solenoid).
Hope this helps!