# A generalization of the Biot-Savart law for a number $n$ of wires with $n\geq 3$

This question is not a homework-exercises, but a request if exist a general formula of Biòt-Savart. We suppose that I have three or more wires traversed by incoming and outgoing currents and I would calculate the resultant of the magnetic field in a generic point of the space.

I have this example where I have a square and in the vertex there are the currents, where $$r$$ it is the half-diagonal of the square.

EDIT: Added new drawing after the comments of the user @G. Smith because the previous drawing it is not correct the direction and the verse of $$\mathbf{B}_{24}$$.

If I wanted to find the summation of the magnetic fields due from the 4 (four) wires, hoping that the drawn vectors of the magnetic fields are drawing correctly (if they are not correct or there is a different drawing, I am very happy to know my mistakes), I should to apply to each pair of wires (for example for $$\mathbf{B}_{13}$$) the relation:

$$\boxed{B_{13}=\frac{\mu_0 (+I_1+I_3)}{2\pi r}}$$ and $$\boxed{B_{24}=\frac{\mu_0 (+I_4-I_2)}{2\pi r}}$$ or $$\boxed{B_{24}=\frac{\mu_0 (-I_4+I_2)}{2\pi r}}$$

Why must I algebraically sum the values of the wire currents? Is there a general formula of Biòt-Savart for this?

• Comments are not for extended discussion; this conversation has been moved to chat. May 8 '20 at 11:53

## 1 Answer

The magnitude of the magnetic field a distance $$\:r\boldsymbol{=}\Vert \mathbf r \Vert\:$$ from a long straight wire carrying a steady current I, see Figure, is given by $$$$\mathrm B\boldsymbol{=}\Vert\mathbf B \Vert\boldsymbol{=}\dfrac{\mu_0}{2\pi}\dfrac{\:\:\rm I\:\:}{r} \tag{01}\label{01}$$$$ while its direction is found by the right-hand rule.

Using the vectors $$\:\mathbf I, \mathbf r\:$$ and $$\:\mathbf B\:$$ in place of the scalar quantities $$\:\mathrm I, r\:$$ and $$\:\mathrm B\:$$ all these informations are contained in the following simple vector equation $$$$\boxed{\:\:\mathbf B \boldsymbol{=}\dfrac{\mu_0}{2\pi}\dfrac{\:\:\mathbf I \boldsymbol{\times}\mathbf r\:\:}{\Vert \mathbf r\Vert^2}\:\:} \tag{02}\label{02}$$$$ This is the Biot-Savart Law for a long straight wire carrying a steady current I.

Note : the OP has confused above Law with its use to find the (Lorentz) force per unit length between two parallel wires carrying parallel or anti-parallel steady currents.

• I see only now your picture. Thank youuuuuuuuuuuu. Jun 8 '20 at 22:14