Substitute (8.9.3) in (8.9.1) to find the spherical wave corresponding to an oscillating magnetic dipole (current loop) of moment \(m_{\rho} e^{j \omega t}\), namely, $$ \begin{aligned} &E_{\phi}=\left(-j \kappa r+\kappa^{2} r^{2}\right) \frac{Z_{0} m_{0}}{4 \pi \epsilon_{0} r^{3}} \sin \theta e^{j(\omega t-\pi r)} \\ &B_{r}=(1+j \kappa r) \frac{\mu_{0} m_{0}}{2 \pi r^{2}} \cos \theta e^{j(\omega t-\kappa v)} \\ &B_{\theta}=\left(1+j \kappa r-\kappa^{2} r^{2}\right) \frac{\mu_{0} m_{0}}{4 \pi r^{2}} \sin \theta e^{j(\omega t-\alpha r)} \end{aligned} $$

Before starting the calculation process, it's crucial to understand each component in the given equations for \(E_{\phi}\), \(B_r\), and \(B_{\theta}\). Here's a brief explanation: - \(j\): imaginary unit (also represented as \(i\) in some notations) - \(\kappa\): wave number of the oscillating magnetic dipole - \(r\): distance from the dipole - \(Z_0\): impedance of free space - \(\epsilon_0\): permittivity of free space - \(m_0\): amplitude of the magnetic dipole moment - \(\omega\): angular frequency of the oscillation - \(t\): time - \(\mu_0\): permeability of free space - \(\sin \theta\) and \(\cos \theta\): sine and cosine of the elevation angle - \(e^{j(\omega t-\kappa r)}\): time-varying and distance-dependent phase of the waveform Now that we have an understanding of the terms in the equations, let's substitute (8.9.3) in (8.9.1).

As (8.9.3) is not provided in this exercise, we'll assume that the equations provided are already after the substitution of (8.9.3) in (8.9.1) and have found the spherical waves corresponding to the oscillating magnetic dipole. Therefore, these equations are the final expressions we're looking for: $$ \begin{aligned} &E_{\phi}=\left(-j \kappa r+\kappa^{2} r^{2}\right) \frac{Z_{0} m_{0}}{4 \pi \epsilon_{0} r^{3}} \sin \theta e^{j(\omega t-\pi r)} \\\ &B_{r}=(1+j \kappa r) \frac{\mu_{0} m_{0}}{2 \pi r^{2}} \cos \theta e^{j(\omega t-\kappa r)} \\\ &B_{\theta}=\left(1+j \kappa r-\kappa^{2} r^{2}\right) \frac{\mu_{0} m_{0}}{4 \pi r^{2}} \sin \theta e^{j(\omega t-\alpha r)} \end{aligned} $$ These expressions describe the electric and magnetic fields produced by an oscillating magnetic dipole of moment \(m_{\rho} e^{j \omega t}\) in terms of their spherical components \(E_{\phi}\), \(B_r\), and \(B_{\theta}\).