XY plane as the interface between two media

[whatisgoingon]

Homework Statement

The xy-plane serves as the interface between two different media. Medium 1 (z < 0) is filled with a material whose µ_r=6, and medium 2 (z > 0) is filled with a material whose µ_r=4. If the interface carries a current (1/µ0)ˆy (y- hat) mA/m, and B2 = 5ˆx (x-hat) + 8ˆz (z-hat) mWb/m2 , find H₁ and B₁.

Homework Equations

H = 1/μ₀ B - M.
H_above - H_below = K_f x ^n (n-hat)[/B]

The Attempt at a Solution

I know I am not doing this right because it seems to simple and my linear algebra isn't the best (I forgot some of the concepts) but this is what made sense to me:

H₂ = 1/(μ₀4) [5 ^x + 8 ^z]

Sub that into the second equation.

so

1/(μ₀4) [5 ^x + 8 ^z] - H₁ = 1 ^y
=>
1/(μ₀4) [5 ^x + 8 ^z] - 1/(μ₀6)B₂ = 1 ^y

I change it into vector coordinates

1/μ₀ (5/4, 0, 8/4) - 1/μ₀ (x/6, -6/6, z/6) = 1 ^y

Then solve for x and z.

I get H₁ = 1/(μ₀) (7.5/6 ^x - 1 ^y + 12/6 ^z)
and B₁ = 7.5 ^x - ^y + 12^zAm I going about this right or am I all the way in Mars right now?

 

[TSny]

Homework Equations

H = 1/μ₀ B - M.
H_above - H_below = K_f x ^n (n-hat)[/B]

The Attempt at a Solution

...
H₂ = 1/(μ₀4) [5 ^x + 8 ^z]

Your result for H₂ is correct, but it isn't clear how you got it. It does not follow immediately from either of your equations in the "Relevant equations" section.

Sub that into the second equation.

so

1/(μ₀4) [5 ^x + 8 ^z] - H₁ = 1 ^y

This isn't right. Your relevant equation $\mathbf H_{\rm above} - \mathbf H_{\rm below} = \mathbf K_f \times \hat {\mathbf n}$ is not correct in general. (Even if it were correct, the right hand side does not have a direction in the $\hat y$ direction.)

The correct equation is generally written as shown here https://en.wikipedia.org/wiki/Inter...terface_conditions_for_magnetic_field_vectors
See the second equation.

 

[whatisgoingon]

Equation 1 of my "Relevant Equations" is how I got H₂. I just inserted B₂ into the equation.

And thank you! That link makes a lot of sense.

I was wondering if I was going in the right direction in terms of subtracting the vectors to get zero. Because from the link you posted, I need to find B₁ and then subtract it from B₂ to get zero? Because from there I can find H₂. Am going about this the right way?

 

[TSny]

Equation 1 of my "Relevant Equations" is how I got H₂. I just inserted B₂ into the equation.

Your equation 1 has the magnetization M. So, it is not immediately obvious how you dealt with this term. After substituting B₂ into the equation, what did you do with M?

I was wondering if I was going in the right direction in terms of subtracting the vectors to get zero. Because from the link you posted, I need to find B₁ and then subtract it from B₂ to get zero? Because from there I can find H₂. Am going about this the right way?

I would need to see the details of your attempt. The equation to use is $\hat {\mathbf n} \times \left( \mathbf H_2 - \mathbf H_1\right) = \mathbf K$, where $\hat {\mathbf n}$ is the unit normal vector pointing from medium 1 to medium 2. Consider separately the x and y components of this equation.