# What does the magnetic force between two current currying wire segments depend on?

The book from which I have taken this picture claims that the force on the segment X in the first case is less than th force acting on it in the second case. I don't think this is true unless one carries out the full calculation. The wire segment on the left causes a magnetic field that changes with each single point along the wire X in each case. So, is there a way in which I can make that comparison without actually currying out the full calculation?

The $$\bf B$$ field for a wire of length L in the x direction is, $$$${\bf B}(x,y)=\frac{{\bf I\times{\hat j}}}{cy}\left[\frac{(L/2-x)}{\sqrt{L/2-x)^2+y^2}}+\frac{(L/2+x)}{\sqrt{L/2+x)^2+y^2}}\right].$$$$ You can derive from this that the field diminishes as x extends beyond L/2. This means there will be a weaker field acting on the second wire when they are displaced. Integrating I$$\bf dl\times B$$ over the second wire also shows that the force is weakens with displacement.
• By $\vec{I}$ do you mean $\vec{I}=IL\hat{n}$
• $\bf I$ is in the x direction, so ${\bf I=}I{\bf\hat i}$ Commented Feb 28 at 19:30
• The center is at the origin. $x$ is the distance from the origin. Commented Feb 29 at 15:28