An isotropic conductor is in the presence of a magnetic induction field B. The electric resistivity rank-2 tensor is a function of the magnetic induction field B and Ohm’s law is written as,∇ϕ = −ρ (B) · jq ,The reversibility of the dynamics at the microscopic scale, implies that the transpose of

Question

An isotropic conductor is in the presence of a magnetic induction field B. The electric resistivity rank-2 tensor is a function of the magnetic induction field B and Ohm’s law is written as,

[latex] abla \varphi = - ho (B) .j_q [/latex].

The reversibility of the dynamics at the microscopic scale, implies that the transpose of the electric resistivity tensor is obtained by reversing the orientation of the magnetic induction field B. Thus,

[latex] ρ^T (B) = ρ (−B) [/latex].

This result cannot be established in a thermodynamic framework but requires the use of a statistical physics. In a linear electromagnetic framework, when the magnetic induction field B is applied orthogonally to the conductive electric current density [latex] j_q [/latex], show that
Ohm’s law can be written as,

[latex] abla \varphi = −ρ · j_q − Hj_q × B [/latex].

where the first term is Ohm’s law (11.74) in the absence of a magnetic induction field B and the second term is the Hall effect (11.75) in a direction that is orthogonal to the magnetic induction field B and to the conductive electric current density. Use the result established in § (11.10).

[latex] abla \varphi = −ρ (s, n_A, q) · j_q [/latex] (11.74)

[latex] abla \varphi =−Η (j_q× B) [/latex] (11.75)

Accepted Answer

We showed in § 11.10 that the electric resistivity tensor ρ (B) can be expressed as the sum of a symmetric part ρs (B) and an antisymmetric part [latex] ρ^α (B) [/latex]. Thus, Ohm’s law (11.74) is recast as,

[latex] abla \varphi = −ρ (s, n_A, q) · j_q [/latex] (11.74)

[latex] abla \varphi = −ρ^s (B) · j_q − ρ^α (B) · j_q [/latex]

The electric resistivity tensor ρ (B) is a linear function of the magnetic induction field B in a linear electromagnetic framework. According to the statistical relation based on the reversibility of the dynamics at the microscopic scale, the electric resistivity tensor is antisymmetric when a magnetic induction field B is present. Thus, the symmetric part of the resistivity tensor [latex] ρ^s = ρ (0) ≡ ρ [/latex] is independent of the magnetic induction field since,

[latex] ρ^T (0) = ρ (0) [/latex]

According to the result established in § (11.10) for the antisymmetric part, Ohm’s law is recast as,

[latex] abla \varphi = −ρ · j_q − ρ^α (B) \hat{u} × j_q [/latex]

where [latex] ρ^α (B) [/latex] is a linear function of the magnetic induction field B and [latex] \hat{u} [/latex] is a dimensionless unit vector that can be chosen orthogonal to the conductive electric current density [latex] j_q [/latex] without imposing restrictions. The anisotropy unit vector ˆu is due to the presence of the magnetic induction field B that breaks the isotropy of the conductor leading to off-diagonal terms in the electric resistivity tensor ρ (B). Therefore, the anisotropy unit vector [latex] \hat{u} [/latex] is oriented along the magnetic induction field B. When the magnetic induction field B is applied orthogonally to the conductive electric current density [latex] j_q [/latex] , the anisotropic term in Ohm’s law can be recast as,

[latex] ρ^α (B) \hat{u} × j_q = −Η B × j_q [/latex]

where [latex] Η = - ho ^\alpha (B)/ \left\|B ight\| [/latex] is a scalar coefficient. Thus, Ohm’s law is recast as,

[latex] abla \varphi = −ρ · j_q − H j_q × B [/latex]

where the first term is Ohm’s law (11.74) and the second term is the Hall effect (11.75).

[latex] abla \varphi = −ρ (s, n_A, q) · j_q [/latex] (11.74)

[latex] abla \varphi =−Η(j_q× B) [/latex] (11.75)

Question 11.11: An isotropic conductor is in the presence of a magnetic indu...

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