Just like there are ways to solve for the force between two straight parallel wires, what is the way we could find the force between non-parallel wires?
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1$\begingroup$ What do you think? Have you tried using the Biot Savart law on this system? Can the wires be taken to be infinite if they are not parallel? $\endgroup$– PhysikerCommented Feb 21 at 7:46
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$\begingroup$ Integrate the force on each infinitesimal segment. $\endgroup$– GhosterCommented Feb 21 at 7:47
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$\begingroup$ Probably 3D is meant so the wires can be infinite and not parallel. Writing down the double integral over the two wires to find the force will be a bit challenging, finding out whether it has a closed-form result would have been the other interesting task in the old days, but nowadays it is of course a matter of "try the engine". $\endgroup$– Jos BergervoetCommented Feb 21 at 7:58
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$\begingroup$ What did you try to answer this question yourself? $\endgroup$– HyperonCommented Feb 21 at 8:05
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$\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$– Community BotCommented Feb 21 at 8:31
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1 Answer
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Assuming we already know how to calculate the $B$-field of one infinite straight wire, we only need the integral of its force over the second wire. Without loss of generality we can assume that the first wire is in the $z$ direction and the second runs through point $(x_c,0,0)$ in the direction ${\bf d}=(0,\sin\alpha,\cos\alpha)$, where $x_c$ is the closest distance between the wires. The field from the first wire is: $$ {\bf B} \sim \frac{1}{(x^2+y^2)^{3/2}} \ (-y,x,0), $$ where computing the front factor is left as an exercise for OP. The force on the second wire is: $$ \int_{-\infty}^{\infty} d\lambda \ {\bf B}(x,y,z) \times {\bf d} $$ where the position is parametrized as $(x,y,z)=(x_c,0,0)+\lambda {\bf d}$. Working this out should be straifghtforward.
(PS: A nice extra problem would be to find the torque of the wires on each other!)
answered Feb 21 at 8:16