Magnetic Circuit with diferent section

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Homework Statement

Given the following magnetic circuit:

It's constituted by two cores in serie. The first one has an average length of 40 cm and the second of 10 cm. Both are made of ferromagnetic material, which can be considered lineal and with [tex]\mu[/tex]_r1 = 1000 for the first one, and [tex]\mu[/tex]_r2 = 2000 for the second one. Additionally S₁= 1 cm², S₂ = 1,5 cm², I = 2A and N = 200.

1) Find the magnetic flux.
2) Find the length of each of the 3 vectors in each material, and indicate their direction.

Homework Equations

1- Boundary conditions: B_1n (the magnetic field in the first core) = B_2n.

2- [tex]\mathop{{\int\!\!\!\!\!\int}\mkern-21mu \bigcirc}\limits_S
{\vec B \cdot d\vec s = 0}[/tex]

3- Extention of Ampère's Law: [tex]\oint\limits_C {\vec H \cdot d\vec l} = NI[/tex]

4- Since we are considering the materials lineal: [tex]\vec B = \mu \vec H[/tex]

The Attempt at a Solution

First of all, I know that because of the canalization effect, B (and therefore H) is going to be as parallel as possible to the circuit. So, except in the corners, B is parallel to the circuit's "walls". I also know that B, H and M outside the circuit are equal to zero.

Well, the first thing I thought was using the extention of Ampère's Law. Since I don't know exactly how to find a curve C that best fits the situation, but I'm given the average length for each core, I guess it doesn't matter. So:

[tex]\oint\limits_C {\vec H \cdot d\vec l = \int\limits_{\mu _1 } {\vec H_1 \cdot d\vec l} } + \int\limits_{\mu _2 } {\vec H_2 \cdot d\vec l} = H_1 .0,4m + H_2 .0,1m = NI[/tex]

Then, because of the boundary condition (2), [tex]B_{\mu _1 n} = B_{\mu _2 n}[/tex]. But because in both boundaries both [tex]B_{\mu _1 }[/tex] and [tex]B_{\mu _2 }[/tex] are normal to the contact surface, then [tex]B_{\mu _1 } = B_{\mu _2 } = B[/tex].

Using the relation (4): [tex]\frac{{B_{\mu _1 } }}{{\mu _1 }} = H_1 [/tex] and [tex]\frac{{B_{\mu _2 } }}{{\mu _2 }} = H_2 [/tex].

Replacing this into the first equation: [tex]\frac{B}{{\mu _1 }}.0,4m + \frac{B}{{\mu _2 }}.0,1m = B\left( {\frac{{0,4m}}{{\mu _1 }} + \frac{{0,1m}}{{\mu _2 }}} \right) = NI[/tex]. Then I find that [tex]B = \frac{{NI}}{{\left( {\frac{{0,4m}}{{\mu _1 }} + \frac{{0,1m}}{{\mu _2 }}} \right)}} = 1,12T[/tex].

The thing is that I think this is wrong, because then I will find one flux for the section S₁ and another for the section S₂. Also, I wouldn't satisfy condition (2).

But if I begin using that condition, I find that [tex]B_{\mu _1 } = B_{\mu _2 } .1,5
[/tex], which violates the boundary condition (1) (MathType doesn't let me copypaste any double integrals, but this comes from calculating the magnetic flux through a close surface S that includes both sections S₁ and S₂).

What am I doing wrong?

 

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Anybody, please?