force is exerted either on a moving or on a stationary charge in the static, or in the time‐varying electric field. The magnetic force is exerted only on a moving charge in the static, or in the time‐varying, magnetic field. In the case of a time‐varying electric, or the time‐varying magnetic field, both fields are always present and are related through Maxwell’s equations (4.4.1a) and (4.4.1b). Thus, both components of Lorentz's force are present on a moving charge in a time‐varying EM‐field.
It is observed that in the absence of the external sources, in a lossless medium Maxwell’s equation (4.4.1a) states that a time‐varying magnetic field creates a time‐varying electric field; and Maxwell’s equation (4.4.1b) states that a time‐varying electric field () creates a time‐varying magnetic field. Thus, Maxwell’s equations (4.4.1a) and (4.4.1b) form a set of the coupled equations, showing an interdependence of the time‐varying electric and magnetic fields. It is like the two‐variable simultaneous equations that occur in ordinary algebra. However, in the present case, the field variables (, are vector quantities. The coupled partial differential equations are solved either for the or by following the rules of the vector algebra. The solutions provide the wave equation either for the electric () or for the magnetic () field.
Equation (4.4.1c) of Maxwell's equations is a divergence relation. It shows that the electric field originates from a charge, and ends on another charge. Equation (4.4.1d) shows that the divergence of the magnetic field is zero; meaning thereby that a magnetic charge does not exist in nature. The magnetic field exists as a closed‐loop around a current‐carrying conductor. However, sometimes a presence of the hypothetical magnetic charge is assumed, and the divergence equation (4.4.1d) is modified as . This assumption maintains the symmetry of Maxwell’s equations. Likewise, to maintain the symmetry of Maxwell’s equations, a hypothetical magnetic current density (−Jm) term is also to be added to equation (4.4.1a). The modified Maxwell equations in the symmetrical form are given below:
(4.4.4)
Figure 4.8 Surfaces and volume used in the integral form of Maxwell equations.
In a real material medium, a current can flow due to the time‐dependent electric polarization (P), and also due to the time‐dependent magnetic polarization (M). These are known as the electric and magnetic polarization currents and are incorporated in Maxwell’s equations as the electric and magnetic displacement current densities. Sometimes, the external sources, magnetic current density and electric current density are further added to Maxwell equations (4.4.1a) and (4.4.1b). The external current sources are retained in the modified form of Maxwell equations. Figure (4.8a) shows externally impressed current sources, creating fields in an enclosure. These externally applied current sources supply power to create the electric field and magnetic field in a material medium. They also cause the flow of current in a lossy medium. It is examined in the next section while discussing the energy balance of the electromagnetic field [B.8, B.9]. The lossless free space is treated as a charge‐free and source‐free medium, i.e. .
The above equation shows the conservation of charge. In any enclosed volume, the decrease of charge is associated with a flow of current out of the volume.
Normally, the time‐harmonic fields are used in most of the applications. The time‐dependent electric field and other field quantities are written in the phasor form for the time‐harmonic field:
(4.4.6)
(4.4.7)
It is noted that the same notation is used for both the time‐space dependent fields [ and only space‐dependent fields [ in the phasor form. The context of the discussion can clarify the situation. In general, the space coordinates based [ etc.] phasor fields are complex quantities, with both the magnitude and phase angle associated with them. The real part gives actual sinusoidal field quantity with phase relation. is the instantaneous field quantity without any phase term. Further, the following relations are useful:
force 