B field due to infinite wire

[cscott]

Homework Statement

I want the force on a finite conducting wire that is perpendicular to an infinite wire.

Can I do it with this:

$$F = I_f \int{\frac{\mu_0 I_i}{2\pi r}}{dr}$$ where $$I_f, I_i$$ are the currents in the finite and infinite wires.

 

[Shooting Star]

Yes, that’s the way to do it. But what is the region of integration? It would be easier to answer if you had described the exact picture you had in mind. Why is Ii inside the integral? I presume you are dealing with steady currents.

Let’s set it up properly. Suppose the infinite wire lies along the y-axis and Ii is toward +ve y-axis. The finite wire lies on the x-axis from x1 to x2 and If is toward the +ve x-axis.

B due to Ii at a pt x on the x-axis =Bi = k*Ii/x, where I’ve written k for mu_0/2pi. Bi points in the –z dircn.

If we consider an elementary length dx at x, then the force on this is dF = If*Bi*(sin 90)dx = k*If*Ii*dx/x

Now you can integrate from x1 to x2 and tell us the magnitude and direction of the force?

 

[ogb p]

Yes, this equation is known as the Biot-Savart law and it can be used to calculate the magnetic field (B-field) at a point due to a current-carrying wire. In this case, you are interested in the force (F) on a finite wire, which can be calculated by integrating the B-field along the length of the wire. The B-field is dependent on the currents in both the finite and infinite wires, as well as the distance (r) between them. It is important to note that this equation assumes the wires are infinitely long and straight, and the finite wire is perpendicular to the infinite wire.