Conducting Wire Homework: Find J0 & B Inside Wire

[dtesselstrom]

Homework Statement

A long, straight conducting wire of radius R has a nonuniform current density J=J0*r/R, where J0 is a constant. The wire carries total current I.
Find an expression for J_0 in terms of I and R.
Find an expression for the magnetic field strength inside the wire at radius r.

Homework Equations

J=I/A
B*ds = u0*I

The Attempt at a Solution

So from my book it looks like J0=I/A which is Jo=I/(pi*R^2) which says it is off by a multiplication factor not sure where I am going wrong here. As for the second portion I got B= uo*I*r/(2*pi*R^2) which is also wrong so any help on this would be much helpful.

 

[robb_]

Since J is a function of r, it will be constant for a thin cylindrical shell centered on the wire. Adding up the current in all these shells will equal I.

 

[m2003]

To find J0, we can use the equation J=I/A, where A is the cross-sectional area of the wire. Since the current density is given as J=J0*r/R, we can substitute this into the equation to get J0=I/(pi*R^2), which is the correct expression for J0 in terms of I and R.

To find the magnetic field strength inside the wire, we can use the equation B*ds = u0*I, where B is the magnetic field strength and ds is the differential length of the wire. Since the wire is straight, we can assume that the magnetic field is also straight. Therefore, we can use the equation B=u0*I/(2*pi*r), where r is the distance from the center of the wire. Substituting the given current density, we get B=u0*J0*r/2, which is the correct expression for the magnetic field strength inside the wire at radius r.