Question

Let \(\mu_{r 1}=2\) in region 1, defined by \(2 x+3 y-4 z>1\), while \(\mu_{r 2}=5\) in region 2 where \(2 x+3 y-4 z<1\). In region 1, \(\mathbf{H}_{1}=50 \mathbf{a}_{x}-30 \mathbf{a}_{y}+\) \(20 \mathbf{a}_{z} \mathrm{~A} / \mathrm{m} .\) Find \((a) \mathbf{H}_{N 1} ;(b) \mathbf{H}_{t 1} ;(c) \mathbf{H}_{22} ;(d) \mathbf{H}_{N 2} ;(e) \theta_{1}\), the angle between \(\mathbf{H}_{1}\) and \(\mathbf{a}_{N 21} ;(f) \theta_{2}\), the angle between \(\mathbf{H}_{2}\) and \(\mathbf{a}_{N 21}\).

Short answer

In this exercise, we are given the magnetic field in region 1, \(\mathbf{H_1}\), and need to determine several components and angles in regions 1 and 2. Follow these steps to find the required values: 1. Find the normal vector to the boundary surface (\(\mathbf{a}_{N21}= 2\mathbf{a}_x + 3\mathbf{a}_y - 4\mathbf{a}_z\)) and its magnitude (\(| \mathbf{a}_{N21} |= \sqrt{29}\)). 2. Find the normal component of \(\mathbf{H_1}\) (\(\mathbf{H}_{N1}\)) and the tangential component (\(\mathbf{H}_{t1}\)). 3. Using the boundary conditions, find \(|H_{N2}|\) and \(\mathbf{H}_{22}\). 4. Calculate the angles \(\theta_1\) and \(\theta_2\) between the magnetic fields and the normal vector using the dot product. By taking these steps, we can find the normal and tangential components of the magnetic field in each region and the angles between the fields and the normal vector.

Step-by-step solution

Find the normal vector

The normal vector to the boundary surface, \(\mathbf{a}_{N21}\), can be found using the coefficients of the given equation: \(2x + 3y - 4z = 1\) \(\mathbf{a}_{N21} = 2\mathbf{a}_x + 3\mathbf{a}_y - 4\mathbf{a}_z\) We can also find the magnitude of the normal vector, \(| \mathbf{a}_{N21} |\). \(| \mathfile{fontsize{12pt}\mid a_{N21} \mid} = \sqrt{2^2+3^2+(-4)^2} = \sqrt{29}\)

Find \(\mathbf{H}_{N1}\) and \(\mathbf{H}_{t1}\)

The normal component of \(\mathbf{H_1}\) is given by the dot product of \(\mathbf{H_1}\) and \(\mathbf{a}_{N21}\) divided by the magnitude of \(\mathbf{a}_{N21}\). \(\mathbf{H}_{N1} = \frac{\mathbf{H_1} \cdot \mathbf{a}_{N21}}{| \mathbf{a}_{N21} |}\) \(\mathbf{H}_{t1} = \mathbf{H_1} - \mathbf{H}_{N1}\)

Find \(\mathbf{H}_{22}\) and \(\mathbf{H}_{N2}\)

Using the boundary conditions on the magnetic field, we have: \(|H_{N1}|\mu_{r1} = |H_{N2}|\mu_{r2}\), so \(|H_{N2}| = \frac{H_{N1}\mu_{r1}}{\mu_{r2}}\) Since \(\mathbf{H}_{t1} = \mathbf{H}_{t2}\), we can find \(\mathbf{H}_{22}\) as follows: \(\mathbf{H}_{22} = \mathbf{H}_{t1} + \mathbf{H}_{N2}\)

Find \(\theta_1\) and \(\theta_2\)

The angles between the magnetic fields and the normal vector can be obtained using the dot product: \(\cos\theta_1 = \frac{\mathbf{H_1}\cdot\mathbf{a}_{N21}}{|\mathbf{H_1}|}\) \(\cos\theta_2 = \frac{\mathbf{H_{22}}\cdot\mathbf{a}_{N21}}{|\mathbf{H_{22}}|}\) Then, we can find \(\theta_1\) and \(\theta_2\) by taking the inverse of the cosine function applied to the found values.

Key concepts

Magnetic Field Boundary Conditions

In magnetostatics, boundary conditions play a crucial role in defining how magnetic fields (\(\mathbf{H}\)) behave at the interface between different materials. Here, we have two regions defined by \(2x + 3y - 4z = 1\), with relative permeabilities \(\mu_{r1}=2\) and \(\mu_{r2}=5\). These permeabilities affect the continuity of the magnetic field components at the boundary.

The normal component of the magnetic field \(H_{N1}\) in region 1 must satisfy the condition:

• \(|H_{N1}|\mu_{r1} = |H_{N2}|\mu_{r2}\)

This equation ensures that the perpendicular component of the magnetic induction \(\mathbf{B}\) remains continuous across the boundary.

The tangential component of the magnetic field \(\mathbf{H}_{t1}\) is continuous:

• \(\mathbf{H}_{t1} = \mathbf{H}_{t2}\)

These conditions help us calculate the field in region 2 once we know \(\mathbf{H}_{1}\)in region 1.

Vector Normalization

Vector normalization is a process of converting a vector into a unit vector that points in the same direction. This operation is vital for simplifying calculations involving vectors, like computing angles or projecting vectors.

For a vector \(\mathbf{a}_{N21}= 2\mathbf{a}_x + 3\mathbf{a}_y - 4\mathbf{a}_z\) which defines the orientation of the boundary,

• Its magnitude is found using: \(\sqrt{2^2 + 3^2 + (-4)^2} = \sqrt{29}\)

The unit vector, or the normalized vector, \(\hat{\mathbf{a}}_{N21}\), is obtained by dividing the vector by its magnitude:

• \(\hat{\mathbf{a}}_{N21} = \frac{1}{\sqrt{29}}(2\mathbf{a}_x + 3\mathbf{a}_y - 4\mathbf{a}_z)\)

Normalization is essential for comparing directions and computing dot products with other vectors.

Dot Product in Electromagnetics

The dot product is a fundamental operation in vector calculus, especially in electromagnetics, for determining the magnitude of a vector component in the direction of another vector. It is used to calculate components like \(\mathbf{H}_{N1}\) from \(\mathbf{H}_{1}\) and the normal vector to a boundary, \(\mathbf{a}_{N21}\).

The dot product is calculated as:

• \(\mathbf{H}_1 \cdot \mathbf{a}_{N21} = (50\mathbf{a}_x - 30\mathbf{a}_y + 20\mathbf{a}_z) \cdot (2\mathbf{a}_x + 3\mathbf{a}_y - 4\mathbf{a}_z) = 100 - 90 - 80\)

This results in the scalar component which, when divided by \(\sqrt{29}\), gives the normal component \(\mathbf{H}_{N1}\).

The dot product helps us find the angle \(\theta_1\)between vectors using:

• \(\cos\theta_1 = \frac{\mathbf{H}_1 \cdot \mathbf{a}_{N21}}{|\mathbf{H}_1|}\)

Understanding the dot product is key to solving problems involving angles and projections of fields on certain surfaces.

Relative Permeability

Relative permeability \(\mu_r\)is a measure of how easily a material can become magnetized relative to vacuum. It is crucial in defining how magnetic fields behave when transitioning between materials. In the exercise, regions have different \(\mu_r\) values:

• Region 1: \(\mu_{r1} = 2\)
• Region 2: \(\mu_{r2} = 5\)

This difference impacts the normal component of the magnetic field at the boundary.

The relationship between \(H_{N1}\) and \(H_{N2}\) is determined by the ratio of these permeabilities:

• \(|H_{N2}| = \frac{H_{N1}\mu_{r1}}{\mu_{r2}}\)

Understanding \(\mu_r\) helps us evaluate material behaviors in magnetic fields. It guides decisions on selecting appropriate materials for specific magnetic field applications, such as shielding or enhancing magnetic effects.