Question

Assuming that the current density \(\mathbf{j}\) and the electric field \(\mathbf{E}\) appearing in equation (21.43) are first-order Cartesian tensors, show explicitly that the electrical conductivity tensor \(\sigma_{i j}\) transforms according to the law appropriate to a second-order tensor. The rate \(W\) at which energy is dissipated per unit volume, as a result of the current flow, is given by \(\mathbf{E} \cdot \mathbf{j}\). Determine the limits between which \(W\) must lie for a given value of \(|\mathbf{E}|\) as the direction of \(\mathbf{E}\) is varied.

Short answer

The conductivity tensor \(\sigma_{ij}\) transforms according to the second-order tensor law. The energy dissipation rate \(W\) varies between the smallest and largest eigenvalues of \(\sigma_{ij}\) times \( |\mathbf{E}|^2\).

Step-by-step solution

- Analyze the given equation

Given equation (21.43) in the problem indicates the relationship between \(\mathbf{j}\), \(\mathbf{E}\), and \(\sigma_{ij}\): \(j_i = \sigma_{ij} E_j\). This expresses the current density \(\mathbf{j}\) as a product of the conductivity tensor \(\sigma_{ij}\) and the electric field \(\mathbf{E}\).

- Understand the transformation properties of tensors

For tensors, first-order tensors transform using linear transformations, while second-order tensors must follow a specific transformation law. A second-order tensor \(T_{ij}\) transforms according to the law: \(T_{ij}' = L_{ik} L_{jl} T_{kl}\), where \(L_{ik}\) and \(L_{jl}\) are the components of the transformation matrix.

- Verify the transformation of \(\sigma_{ij}\)

To show that \(\sigma_{ij}\) transforms like a second-order tensor, consider the transformations under rotation: \(j_i' = L_{ij} j_j\) and \(E_i' = L_{ij} E_j\). Substituting these transformations into the original equation \(j_i = \sigma_{ij} E_j\) transforms accordingly: \(L_{ik} j_k = \sigma_{ij} L_{jl} E_l\). This confirms that \(\sigma_{ij}\) transforms according to the second-order tensor transformation law.

- Determine the energy dissipation rate

The energy dissipation rate per unit volume is given by the dot product \(W = \mathbf{E} \cdot \mathbf{j} = E_i j_i\). Using the relationship \(j_i = \sigma_{ij} E_j\), we get \(W = E_i \sigma_{ij} E_j\). This expression quantifies the energy dissipation for a particular value of the electric field \(\mathbf{E}\).

- Evaluate the limits of \(W\) for given \( |\mathbf{E}|\)

Assure \(\mathbf{E}\) varies in all possible directions while its magnitude \( |\mathbf{E}| \) is constant. The energy dissipation rate \(W = E_i \sigma_{ij} E_j\) must lie between the minimum and maximum eigenvalues of the \(\sigma_{ij}\) tensor, modified by \( |\mathbf{E}|^2\).

Key concepts

current density

Current density, denoted by the symbol \(\mathbf{j}\), is a measure of the electrical current per unit area of cross-section. It indicates how much electric charge is flowing through a specific area. Imagine it as the flow of water through a pipe - current density tells us how much water passes through a segment of the pipe.

electric field

The electric field, represented by \(\mathbf{E}\), is a vector quantity that describes the force exerted by electric charges in space. In simple terms, it tells us how an electric charge would move if placed in a specific location. If we place a positive charge in an electric field, it will experience a force in the direction of the field lines. The strength and direction of the electric field are crucial in determining how charged particles will behave.

second-order tensor

A second-order tensor, like the electrical conductivity tensor \(\sigma_{ij}\), is a mathematical object that provides a more complex level of description than vectors and scalars. While scalars have magnitude and vectors have magnitude and direction, second-order tensors have multiple components and can describe how quantities change in different directions.