Magnetic Field in a cilindrical conductor

[Jaller1404]

Homework Statement

Hey guys, I'm having trouble with a question covering magnectic field on a cilindrical conductor with a cilindrical cavity, here's the question:

Consider a long cilindrical conductor with radius R1, with a cilindric cavity with radius a, in which R1>a. The axis of both cilinders are paralel, and are separated by a distance D≤R1-a. Assume that there's a uniform current density (J₀) flowing through the cilinder's axis. Applying Ampere's law, use the concept of superposition of fields to calculate the resulting magnetic field on points

a) inside the cavity
b) outside the cavity but inside the conductor
c) outside the conductor

Homework Equations

Ampere's Law

B=(μ₀*i)/2πR

The Attempt at a Solution

The attempt is attached below. But I couldn't figure out if the field inside the cavity is the field created by the surroundings of it or the field created by the whole conductor and avaliated inside the cavity. If it's the second option, how can I define it there?

 

[nilesh_pat]

a) Inside the cavityThe magnetic field inside the cavity will be zero. This is because, by symmetry, all the currents that pass on the sides of the cylinder will cancel each other out, and there won't be any net current within the cavity. b) Outside the cavity but inside the conductorThe magnetic field outside the cavity but still inside the conductor will be equal to the field generated by the current density J0 flowing through the cylinder's axis. This can be calculated with the equation B=(μ0*i)/2πR, where i is equal to J0*π(R1^2 - a^2), as this is the total current that passes through the cylinder's axis.c) Outside the conductorOutside the conductor, the magnetic field will be zero. This is because, due to the long cylindrical conductor, the magnetic field created by the current density J0 will decrease rapidly as we move away from the conductor, and thus, will be zero at points far away from the conductor.