Magnetic field vector due to linear conductor

[cdummie]

Homework Statement

Through linear conductor flows current I, with direction shown in the picture. Axis where conductor is placed is common edge of three areas with different ferromagnetic materials. They form angles θ₁, θ₂, θ₃ (θ₁ + θ₂ + θ₃ = 2π). If space is filled with homogeneous materials with permeabilities μ₁, μ₂, μ₃ respectively, find intensity of magnetic field vector for arbitrary point in the space due to linear conductor.

Homework Equations

Generalized Ampere's law.

The Attempt at a Solution

This is my attempt:

Considering the direction of the current, it's obvious that intensity of magnetic field vector depends on the distance from the linear conductor. Since the vector B is normal to the boundary of two materials, from the boundary conditions we have that B is same for any point with same distance from the conductor regardless of the material, which means that H is different in different materials.

Using generalized Ampere's law we have:

∫H*dl=∫H*dl*cos(H,dl)= H₁*θ₁*r + H₂*θ₂*r + H₃*θ₃*r

∑I=IIf we express H₂ and H₃ as B/μ₂ and B/μ₃ respectively we get the expression for H₁ and we can do similar thing for H₂ and H₃.

so,

H₁=(I-B*r(θ₂/μ₂ + θ₃/μ₃))/θ₁*r

Is this correct, and is this reasonable approach?

 

[mfb]

You still have H₁ and B, one of them has to disappear for the final result.

The approach looks fine so far.

 

[cdummie]

You still have H₁ and B, one of them has to disappear for the final result.

The approach looks fine so far.

I know what you mean, i forgot to express B in final formula, which i would do by expressing H₁, H₂ and H₃ in first formula using B divided by corresponding permeability that way i could easily find B since it's the same for any point regardless of material.

 

[rude man]

Not sure what the OP said in post 3, but the expression for H_i (or B_i) should have nothing but I, θ₁, θ₂, θ₃ and r in it. No H's or B's. If this is what the OP meant I have no additioal comment.

 

[cdummie]

Not sure what the OP said in post 3, but the expression for H_i (or B_i) should have nothing but I, θ₁, θ₂, θ₃ and r in it. No H's or B's. If this is what the OP meant I have no additioal comment.

I wasn't clear enough, sorry, anyway, this is what i meant:

 

[rude man]

I wasn't clear enough, sorry, anyway, this is what i meant:

View attachment 88740

Fine! I thought that's what you meant but wasn't sure.

 

[mfb]

The last expression for H₁ can be simplified significantly (e.g. starting from the expression for B).

 

[cdummie]

The last expression for H₁ can be simplified significantly (e.g. starting from the expression for B).

I know, i should get much "smaller" expression, i could do that, but the most important thing for me was to see if i solved it correctly.