# Force between two current carrying wires: the general case

Assume two straight current carrying parallel wires with currents ($I$ and $I'$) flowing in the same direction, at a distance $R$ from each other. From Ampère's law (and from Biot-Savart as well) it follows that the magnetic field at a distance $R$ induced by the current $I$ is the following. $$\mathbf{B} = \frac{\mu_0 I}{2\pi R}\mathbf{e}_{B}$$ The force on the current $I'$ due to the magnetic field induced by the current $I$ will be the following. $$\mathbf{F}' = \int I\mathbf{e}_{I'} \times \mathbf{B}dl'$$ This follows from the fact that in general $\mathbf{F} = \int \mathbf{j} \times \mathbf{B}dV$ where $dV$ is a segment of a volume V, which in turn follows from the Lorentz force. Substituting $\mathbf{B}$ due to $I$ into the the equation for $\mathbf{F}'$ we get that. $$\mathbf{F}' = \int \mathbf{e}_{I'} \times \mathbf{e}_{B}\frac{\mu_0 II'}{2\pi R}dl' = \int -\mathbf{e}_R\frac{\mu_0 II'}{2\pi R}dl'$$ And as will be recognized the integrand is a constant over the whole wire and therefore it follows that. $$\mathbf{F}' = -\mathbf{e}_r\frac{\mu_0 II'}{2\pi R}L'$$ The fact that $\mathbf{e}_{I'} \times \mathbf{e}_{B} = -\mathbf{e}_R$ follows from our choice of origin (see picture below).

If we do the same calculation for $\mathbf{F}$ then we will see that the two wires attract each other. If we do the same calculation with currents in opposite directions we will see they repel.

This is how I learned the derivation but in the textbook I got it from (Alonso & Finn: Volume II: electromagnetism) they say this result can be generalized for all wires, meaning they can have any form and don't necessarily have to be straight. They leave it up to the reader to find the derivation for the general case. My question is simple: what is the derivation for the general case? And does there exist a general case regarding currents only, meaning regardless of the conductor carrying the current?

I have searched for the general case on physics.stackexchange but could only find questions regarding specific problems with forces between current carrying wires not the derivation I was looking for. My apologies if I did not search well enough.

• If they don't have to be straight, then $d$ is not well defined. What exactly does Alonso and Finn say? Do they give a result without a derivation? How general is their "general case"? What do you mean by "regardless of the conductor"? I don't know what you are asking ... – garyp Apr 6 '15 at 15:03
• They say (and I quote): "This result can be generalized for conductors of all forms. The reader can proof for himself that the two conductors in fig 2.31a attract each other..." (fig 2.31a is a picture of two random wires (they are both the same) with currents flowing in the same direction.) They do not give a result and by 'regardless of the conductor' I mean that the conductor does not necessarily have to be a wire. – LucasRouckhout Apr 6 '15 at 15:16
• OK. Well, I still don't know what generalization they mean. Two conductors of any sort will attract each other if their currents go in more or less the same direction ... but that's not much of a leap from the infinite straight wire case. – garyp Apr 6 '15 at 18:57