Electric Field Components in Charge-Free Space

The electric field components in a region of space are given by the expression:

$E_i = a_i + \sum_{j=1,2,3} b_{ij} x_j$

Here, $E_i$ represents the components of the electric field $\mathbf{E}$, $x_j$ are the spatial coordinates ($x_1, x_2, x_3$), and $a_i$ and $b_{ij}$ are constants. The region is stated to be devoid of any charge or current sources.

A region devoid of charge ($\rho=0$) and current ($\mathbf{J}=0$) implies that the electric field $\mathbf{E}$ must satisfy Maxwell's equations in free space. Assuming the field is time-independent (electrostatic case), the relevant equations are:

Maxwell's Equations for Electric Field

• Divergence of $\mathbf{E}$: $\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0} = 0$
• Curl of $\mathbf{E}$: $\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} = 0$ (for static fields)

We will use these two conditions to find the constraints on the constants $b_{ij}$. The matrix $b_{ij}$ has $3 \times 3 = 9$ components in general.

Divergence Constraint on $b_{ij}$ Matrix

The divergence of the electric field is the sum of the partial derivatives of its components with respect to the corresponding coordinates:

$\nabla \cdot \mathbf{E} = \frac{\partial E_1}{\partial x_1} + \frac{\partial E_2}{\partial x_2} + \frac{\partial E_3}{\partial x_3}$

Let's calculate the partial derivatives using the given expression for $E_i$:

• $\frac{\partial E_1}{\partial x_1} = \frac{\partial}{\partial x_1} (a_1 + b_{11}x_1 + b_{12}x_2 + b_{13}x_3) = b_{11}$
• $\frac{\partial E_2}{\partial x_2} = \frac{\partial}{\partial x_2} (a_2 + b_{21}x_1 + b_{22}x_2 + b_{23}x_3) = b_{22}$
• $\frac{\partial E_3}{\partial x_3} = \frac{\partial}{\partial x_3} (a_3 + b_{31}x_1 + b_{32}x_2 + b_{33}x_3) = b_{33}$

Applying the divergence condition $\nabla \cdot \mathbf{E} = 0$:

$b_{11} + b_{22} + b_{33} = 0$

This equation represents a linear constraint on the diagonal components of the matrix $b_{ij}$. It means the trace of the matrix $b_{ij}$ must be zero.

Curl Constraint on $b_{ij}$ Matrix

The curl of the electric field is given by:

$\nabla \times \mathbf{E} = \left( \frac{\partial E_3}{\partial x_2} - \frac{\partial E_2}{\partial x_3} \right)\hat{i} + \left( \frac{\partial E_1}{\partial x_3} - \frac{\partial E_3}{\partial x_1} \right)\hat{j} + \left( \frac{\partial E_2}{\partial x_1} - \frac{\partial E_1}{\partial x_2} \right)\hat{k}$

Applying the curl condition $\nabla \times \mathbf{E} = 0$, each component of the curl must be zero. Let's calculate the required partial derivatives:

• $\frac{\partial E_3}{\partial x_2} = b_{32}$, $\frac{\partial E_2}{\partial x_3} = b_{23}$. Condition: $b_{32} - b_{23} = 0 \implies b_{32} = b_{23}$
• $\frac{\partial E_1}{\partial x_3} = b_{13}$, $\frac{\partial E_3}{\partial x_1} = b_{31}$. Condition: $b_{13} - b_{31} = 0 \implies b_{13} = b_{31}$
• $\frac{\partial E_2}{\partial x_1} = b_{21}$, $\frac{\partial E_1}{\partial x_2} = b_{12}$. Condition: $b_{21} - b_{12} = 0 \implies b_{21} = b_{12}$

These conditions ($b_{ij} = b_{ji}$ for $i \neq j$) imply that the matrix $b_{ij}$ must be symmetric.

Matrix $b_{ij}$ Properties

Based on Maxwell's equations for an electric field in a charge and current-free region, the matrix of constants $b_{ij}$ must satisfy two key properties:

• It is symmetric: $b_{ij} = b_{ji}$ for all $i, j$.
• It is traceless: $\mathrm{Tr}(b) = b_{11} + b_{22} + b_{33} = 0$.

Independent Components Count

A general $3 \times 3$ matrix has 9 independent components. The derived properties impose constraints that reduce this number.

First, consider the symmetry constraint. For a $3 \times 3$ matrix, the symmetry $b_{ij} = b_{ji}$ implies:

The components are determined by the 3 diagonal elements ($b_{11}, b_{22}, b_{33}$) and the 3 unique off-diagonal elements ($b_{12}, b_{13}, b_{23}$). The lower triangle components are fixed by the upper triangle ones. So, a symmetric $3 \times 3$ matrix has $3 + 3 = 6$ independent components.

Second, consider the traceless constraint ($b_{11} + b_{22} + b_{33} = 0$). This condition imposes one linear relationship among the three diagonal components.

For example, if we choose $b_{11}$ and $b_{22}$ independently, then $b_{33}$ is determined by $b_{33} = -(b_{11} + b_{22})$. This constraint reduces the number of independent diagonal components from 3 to 2.

The number of independent components is the number of independent components in a symmetric $3 \times 3$ matrix (6) minus the number of additional constraints (1) from being traceless.

Total independent components = $6 - 1 = 5$.

The five independent components can be chosen as, for example, $b_{11}, b_{22}, b_{12}, b_{13}, b_{23}$. The remaining components are then determined by the constraints.

Thus, there are 5 independent components of the matrix $b_{ij}$.