Consider \(\mathbf{E}\) and \(\mathbf{B}\) wave fields whose only dependence on \(z\) and \(t\) is included in the factor \(e^{i\left(\omega t-x_{1} \theta\right)}\). Further assume TE waves such that \(E_{z}=0\). Write out Maxwell's curl equations \((82.2)\) and \((8.2 .4)\) in cartesian components and show \((a)\) that all four transverse field components can be obtained from \(B_{t}\) by first-order partial differentiation and \((b)\) that \(B_{*}\) must be a solution of the Helmholtz equation \((8.7 .16)\). Thus the scalar function \(\phi\) of the text may be interpreted as proportional to \(B_{z}\) for TE waves or proportional to \(E_{s}\) for TM waves.

The given dependence on \(z\) and \(t\) is included in the factor \(e^{i(\omega t-x_{1}\theta)}\). So, we can represent the electric and magnetic fields as follows: \(\mathbf{E} = \mathbf{E}_{t}(x,y)e^{i(\omega t-x_{1}\theta)}\) \(\mathbf{B} = \mathbf{B}_{t}(x,y)e^{i(\omega t-x_{1}\theta)}\) Where \(\mathbf{E}_{t}(x,y)\) and \(\mathbf{B}_{t}(x,y)\) are the transverse components of the electric and magnetic fields.

Maxwell's curl equations are: (8.2.2) \(\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}\) (8.2.4) \(\nabla \times \mathbf{B} = \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t}\) We need to write these equations in cartesian components. Since we are dealing with TE waves, \(E_z=0\). Therefore, the equations become: \(\frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y} = -\frac{\partial B_z}{\partial t}\) \(\frac{\partial B_z}{\partial x} - \frac{\partial B_x}{\partial z} = \mu_0 \varepsilon_0 \frac{\partial E_y}{\partial t}\) \(\frac{\partial B_y}{\partial z} - \frac{\partial B_z}{\partial y} = \mu_0 \varepsilon_0 \frac{\partial E_x}{\partial t}\) Now, we will use these equations to prove the required conditions for both (a) and (b).

Using the given dependence on \(z\) and \(t\) for \(\mathbf{E}\) and \(\mathbf{B}\), we can differentiate Maxwell's curl equations in Cartesian components with respect to \(t\) and \(z\). We get: (1) \(\frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y} = -i\omega B_z e^{i(\omega t-x_{1}\theta)}\) (2) \(\frac{\partial B_z}{\partial x} - \frac{\partial B_x}{\partial z} = -i x_{1}\theta E_y e^{i(\omega t-x_{1}\theta)}\) (3) \(\frac{\partial B_y}{\partial z} - \frac{\partial B_z}{\partial y} = -i x_{1}\theta E_x e^{i(\omega t-x_{1}\theta)}\) We can see from equations (1), (2), and (3) that all four transverse field components (\(E_y\), \(E_x\), \(B_y\), and \(B_x\)) can be obtained from the transverse field component \(B_{t}\) by first-order partial differentiation with respect to \(x\), \(y\), and \(z\).

The Helmholtz equation is given by: (8.7.16) \((\nabla^2 + k^2)B_{*} = 0\) where \(k=\omega\sqrt{\mu_0 \varepsilon_0}\) is the wave number. To show that \(B_{*}\) must be a solution of the Helmholtz equation, we need to substitute the given dependence on \(z\), \(t\), and use Maxwell's curl equations. Using the curl of (8.2.4) equation, we get: \(\nabla \times (\nabla \times \mathbf{B}) = \nabla \times (\mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t})\) Simplifying and writing in cartesian components, we get: (4) \(\nabla^2 B_z = \mu_0 \varepsilon_0 \frac{\partial^2 B_z}{\partial t^2}\) Now we substitute the given dependence on \(z\) and \(t\) in equation (4): \((\nabla^2 + k^2) B_z e^{i(\omega t-x_{1}\theta)} = 0\) This implies that the \(B_{*}\) must be a solution of the Helmholtz equation \((8.7.16)\). In conclusion, we have shown that all four transverse field components can be obtained from \(B_{t}\) by first-order partial differentiation and that \(B_{*}\) must be a solution of the Helmholtz equation. This means that the scalar function \(\phi\) in the text may be interpreted as proportional to \(B_{z}\) for TE waves or proportional to \(E_{s}\) for TM waves.