Question

The x=0 plane is at the interface between air and a magnetic material with μr=10. The magnetic-field intensity H in air (x<0) along the entire x=0 plane is:

Hₓ₀,air=10A/mx+3A/my−7A/mz
Assuming there is no surface current, find the H-field in the magnetic material (x> 0 ) close to the interface.

Answer 1

The tangential component of the magnetic field intensity in the magnetic material close to the interface is
`\( 10 \, \text{A/m} \) in the \( x \)-direction.`

Step-by-step explanation:

To find the magnetic field intensity
`(\(H\))` in the magnetic material (x > 0) close to the interface, we can use the boundary conditions for magnetic fields at an interface between different materials.

The boundary condition for the magnetic field intensity at the interface is given by:

`\[ H_(t1) - H_(t2) = K_s * \hat{n} \]`

where:

`\( H_(t1) \)` and
`\( H_(t2) \)` are the tangential components of
`\( H \)` on either side of the interface,

`\( K_s \)` is the surface current density,

`\( \hat{n} \)` is the unit normal vector pointing from material 1 to material 2.

Since there is no surface current
`(\( K_s = 0 \))`, the equation simplifies to:

`\[ H_(t1) = H_(t2) \]`

Now, let's consider the given magnetic field intensity in air
`(\( x < 0 \)):\[ \mathbf{H}_{\text{air}} = 10 \, \text{A/m} \cdot \mathbf{x} + 3 \, \text{A/m} \cdot \mathbf{y} - 7 \, \text{A/m} \cdot \mathbf{z} \]`

The tangential component in the air side
`(\( H_(t1) \))` is the component parallel to the interface, which is the
`\( x \)-component` in this case:

`\[ H_(t1) = 10 \, \text{A/m} \]`

Now, since there is no surface current,
`\( H_(t1) = H_(t2) \)`. Therefore, the magnetic field intensity in the magnetic material close to the interface
`(\( x > 0 \))` is also:

`\[ H_(t2) = 10 \, \text{A/m} \]`

So, the tangential component of the magnetic field intensity in the magnetic material close to the interface is
`\( 10 \, \text{A/m} \) in the \( x \)-direction.`