A dielectric material is characterized by a matrix (tensor) permittivity 7 3 -20.5 - 20.5 10...

 

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A dielectric material is characterized by a matrix (tensor) permittivity 7 3 -2√0.5 - 2√0.5 10 -2√0.5 €xx Exy Exy Eyy tyz = Exz €yz € zz when referred to the xyz coordinate frame. If the coordinate axis is rotated into the principal axis u, u, w, the permittivity is exhibited in diagonal form: Dy D. €0 4 €0 [€] = 7 3 -2√0.5 Find the principal axis and the values of the principal dielectric constants €uu/€0, etc. Hint: By definition, along a principal axis a scalar equation such as Eu Eu holds. In general, if D is directed along a principal axis, then Du 7-4A/0 3 -2√0.5 €ul 0 0 0 0 0 €ww € Uu 0 3 7 -2√0.5 3 7 -2√0.5 + or in words, when D is directed along a principal axis, it is related to E by a scalar constant A. The above constitutes a set of three homogeneous equations, of which the first is -2√/0.5 Ex -2√0.5 E, A 10 7€0 2€ √0.5 4 -E₂-0 4 Verify that, for a solution, the following determinant must vanish: 3 7-4A/€0 - 2√0.5 -2√0.5 - 2√0.5 = 0 104A/0 E Answer: € = 30, u = 2€0, €ww = €0. Unit vectors along the principal axis are This cubic equation gives three roots for A, which may be identified as Euur Ev €ww. For any one root, say the components of a vector directed along the corresponding principal axis are proportional to the cofactors of the above determinant. This type of problem is called a matrix eigenvalue problem. The A's are the eigenvalues. √0.5 a, a₁ = 0.5a, + 0.5a, a=0.5a, + 0.5a, + √0.5 a, a = √0.5 a, - √0.5 a, A dielectric material is characterized by a matrix (tensor) permittivity 7 3 -2√0.5 - 2√0.5 10 -2√0.5 €xx Exy Exy Eyy tyz = Exz €yz € zz when referred to the xyz coordinate frame. If the coordinate axis is rotated into the principal axis u, u, w, the permittivity is exhibited in diagonal form: Dy D. €0 4 €0 [€] = 7 3 -2√0.5 Find the principal axis and the values of the principal dielectric constants €uu/€0, etc. Hint: By definition, along a principal axis a scalar equation such as Eu Eu holds. In general, if D is directed along a principal axis, then Du 7-4A/0 3 -2√0.5 €ul 0 0 0 0 0 €ww € Uu 0 3 7 -2√0.5 3 7 -2√0.5 + or in words, when D is directed along a principal axis, it is related to E by a scalar constant A. The above constitutes a set of three homogeneous equations, of which the first is -2√/0.5 Ex -2√0.5 E, A 10 7€0 2€ √0.5 4 -E₂-0 4 Verify that, for a solution, the following determinant must vanish: 3 7-4A/€0 - 2√0.5 -2√0.5 - 2√0.5 = 0 104A/0 E Answer: € = 30, u = 2€0, €ww = €0. Unit vectors along the principal axis are This cubic equation gives three roots for A, which may be identified as Euur Ev €ww. For any one root, say the components of a vector directed along the corresponding principal axis are proportional to the cofactors of the above determinant. This type of problem is called a matrix eigenvalue problem. The A's are the eigenvalues. √0.5 a, a₁ = 0.5a, + 0.5a, a=0.5a, + 0.5a, + √0.5 a, a = √0.5 a, - √0.5 a,

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Okay here are the steps to solve this problem 1 Write the eigen equation E I 0 Where E is t... View the full answer