Magnetic field inside the cavity of a long straight wire

[Irrelativity]

Homework Statement

A long straight wire of radius R carrying a current I has a circular portion of radius a cut out at a distance d from the center, as shown in the picture. Find the magnetic field inside the cut-out portion.

Homework Equations

magnetic field equation (I can't write it here)

The Attempt at a Solution

I couldn't integrate the magnetic field directly. So I tried many tricks to solve this problem. I tried to reduce this into two different straight wires and worked out geometry but couldn't figure it out. If there is no cavity, it's very simple problem. But since I need to find the magnetic field inside the cavity, I need something else.

 

[marcusl]

Start by expressing the field a distance r from the axis of a wire carrying a uniform current density J. You can do this by applying Ampere's law to a circle of radius r. (I assume from your comments that you've done this already.)

You can make a cavity by superposing a second wire of radius a carrying -J, offset by d. Solve for the field at that point r and you should have your answer. You'll find that the field in the hole has a special property.

 

[Irrelativity]

Start by expressing the field a distance r from the axis of a wire carrying a uniform current density J. You can do this by applying Ampere's law to a circle of radius r. (I assume from your comments that you've done this already.)

You can make a cavity by superposing a second wire of radius a carrying -J, offset by d. Solve for the field at that point r and you should have your answer. You'll find that the field in the hole has a special property.

Do I have to assume that the current is steady or varies within the wire? The problem did not state that the current is steady (looks like it really doesn't matter but just in case). And when you find the magnetic field inside the cavity, you do have to take the vector components into the account? thanks.

 

[marcusl]

Do I have to assume that the current is steady or varies within the wire? The problem did not state that the current is steady (looks like it really doesn't matter but just in case). And when you find the magnetic field inside the cavity, you do have to take the vector components into the account? thanks.

Your question is worded badly. Do you mean steady as in non-time-varying? Doesn't matter. Spatially uniform? Well, you tell me.

Vector components? Of course they matter.

 

[Irrelativity]

Your question is worded badly. Do you mean steady as in non-time-varying? Doesn't matter. Spatially uniform? Well, you tell me.

Vector components? Of course they matter.

What about variable r? is r from the center of the wire or from the center of the cavity? right now I am working on r from the center of the cavity.