Magnetic force in the inside of cylindrical conductor?

Question

I'm trying to solve a problem and I can't actually understand what I need to find. I'll try translating in English hoping it will make some sense.

We have a cylindrical conductor carrying a uniformly distributed current I. If μ is the permeability of the conductor's material, find the pressure due to the magnetic field of the conductor at a distance r from its center (inside the conductor).

It's one of my first problems in electromagnetism and I might ask things that are obvious.Here's what I tried:

I know that if r < R (radius of the conductor) then B is given by:

$$ B=\frac{μIr}{2πR^2} $$ I proved this using Ampere's Law. Now I want to find the force. I know it is given by this equation: $$\vec F=q(\vec u\times\vec B)$$ The notes I follow use the other formula $F=BIL$ but I have trouble using any of them. Can I use the second one here? What is my L? And what is meant by pressure at a certain distance? Shouldn't I have an area to find the pressure?

I can write I' in terms of r: $I'=Jπr^2=Iπr^2/πR^2$

I also have B in terms of r already. How do I continue? And how do I make dr show up when I take $d\vec F$ ? That's something I always confuse. If I replace everything I will have r but not dr. How will I integrate afterwards?

Answer 1

You can use the formula for magnetic pressure, $$P=\frac{B^2}{2\mu}=\frac{1}{2\mu}(\frac{\mu I r}{2\pi R^2})^2=\frac{\mu I^2r^2}{8\pi^2 R^4}$$