How to Determine the Force on a Conductor in a Magnetic Field?

[cdummie]

Homework Statement

Say we have a very long conductor shaped like half circular cylinder . With radius a and negligible thickness with constant current through it . At the axis of the cylinder, there is linear conductor with constant current through it . If these conductors are placed in vacuum, determine the force action on the linear conductor.

Homework Equations

The Attempt at a Solution

Since i have to know magnetic field vector (B) so i could find force, i first calculated it, doing it this way, i considered that half cylinder consists of huge number of linear conductors, and i got this for the one linear conductor: B=(μ₀*I)/2πa , since this is just a piece of magnetic field i labeled it dB, and the current dI,
considering dI/dl=I/aπ (it's dI over dL) i got dI=(IdL)/aπ. Now, after i considered x and y components of the vector and calculated it's values i got B_x=-(μ₀I)/aπ² and the B_y was zero. Now, what confuses me is while calculating the value of the one linear conductor I'm not sure if i did it correct, i mean it's basically calculating the magnetic field of one very long linear conductor to another very long linear conductor, I'm not sure can i do it this way

 

[Hesch]

It seems that you are using Ampere's law. But since the shape of the conductor is not symmetrical (half a cylinder), you must use the Biot-Savart law (google).

Find the B-field by (numerical) integration over the half cylinder.

 

[cdummie]

It seems that you are using Ampere's law. But since the shape of the conductor is not symmetrical (half a cylinder), you must use the Biot-Savart law (google).

Find the B-field by (numerical) integration over the half cylinder.

I actually used Biot-Savart law, i just can't write whole derivation, but that's not the problem, i don't understand what would be the difference if i had just a single point instead of whole linear conductor. Because, the way i did this is the same way i would do it if i had just single point

 

[Hesch]

Using Biot-Savart you must integrate over the volume of the conductor: Say you place the conductor in at system of co-ordinates with the center of the cylinder along the x-axis. You will find the B-field at (x,y,z) = (0,0,0). Of cause at tiny element of conductor at (x,y,z) = (0,1,1) will influence the B-field at (0,0,0), but so will a tiny element at (1,1,1). ( A tiny element at (1,0,0) will not ).

That's what Biot-Savart states: You must integrate over all tiny elements.

Therfore you will also have to integrate along the conductor (x-direction).

 

[rude man]

Consider a differentially thin strip of length L along the half-cylinder. You learned the formula for force between two current-carrying wires. Compute the differential force on the inner wire due to this thin strip. Take advantage of any symmetry between one-half of the cylinder and the other half; you might be able to work in one coordinate only. Then integrate over the total half-cylindrical surface. This is not a volume integral problem. The answer is a very simple term in current i (and 2 constants).