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4.4.1 Maxwell’s Equations
ОглавлениеThe set of Maxwell’s equations, given below, consists of four time‐dependent vector equations; relating the sources such as conduction current density (), electric displacement current density , and magnetic displacement current density to the magnetic field ( and the electric field ( respectively:
All quantities in the above equations are space‐time dependent. The current densities , , and are not the power supplying sources to the propagating EM‐wave in a medium. These current densities are created by the externally applied magnetic and electric current densities and supplying power to the EM‐wave and partly getting absorbed as loss in a conducting medium.
The right‐hand sides of the above equations could be treated as the sources (excitations) and the left‐hand fields as the responses. The force field quantities (,), flux field quantities (), and current densities are functions of both the space variables and time variable. Further, in any material medium, the flux field quantities are related to the force field quantities by the constitutive relations, given in equation (4.1.7). The conduction current density in a lossy medium is related to the electric field, as given in equation (4.1.9).
In general, εr, μr, σ are the tensors quantities for an anisotropic medium. However, these are scalar quantities for an isotropic medium. They are also treated as complex quantities to include the losses of a medium. In the case of a complex conductivity σ*, its real part is responsible for the loss in a medium, whereas its imaginary part accounts for the energy storage. In a dispersive medium, εr, μr and σ are also frequency‐dependent. The characteristics of various kinds of media, such as dielectrics, conductors, plasma, semiconductors, ferrites, and so forth are accounted for in Maxwell’s equations through the constitutive relations applicable to these physical media. Maxwell’s equations, along with the constitutive relations, are the field equations, not the force equations, i.e. these equations do not express the forces exerted by the fields on stationary or moving charges. This is achieved through Lorentz’s force equation:
(4.4.2)
where q is the charge on a mass m that is moving with velocity v. In this case, Lorentz's force acting on a moving charge is equal to Newtonian force:
(4.4.3)
Lorentz's force has two components: (i) electric force component and (ii) magnetic force component. The electric 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. .
A divergence of Maxwell’s equations (4.4.1b) in absence of and using equation (4.4.1d), leads to the following continuity equation:
(4.4.5)
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:
(4.4.8)
Using the above equation for the electric field and similar equation for the magnetic field, Maxwell’s equations (4.4.1), in the external source free medium, are rewritten below in the phasor form for the time‐harmonic field:
where flux density vectors are related to field intensity vectors in an anisotropic medium by the following tensor constitutive relations:
(4.4.10)
In the case of an external source‐free homogeneous isotropic medium, Maxwell’s equations are written in terms of field intensities only:
where the scalar constitutive relations are , , . The differential form of the above given Maxwell’s equations provide relations between the electric field and the magnetic field at any location in the medium. Also, the sources are specified at a point in the medium. The Maxwell equations (4.4.9) involving the flux densities and field intensities apply to both the isotropic and anisotropic media, whereas Maxwell’s equations (4.4.11) involving the field intensities apply to the isotropic medium only.
The differential form of Maxwell equations does not account for the creation of the fields in the space due to the sources such as the charge or current distributed over a line, surface, or volume. This case is incorporated in Maxwell’s equations by converting them to the integral forms. It is achieved with the help of two vector identities:
Figure (4.8b) shows the existence of a vector over the surface S. Its boundary is enclosed by the perimeter C. Stoke theorem is defined with respect to Fig. (4.8b) and Gauss divergence theorem with respect to Fig. (4.8c). The unit vector shows the direction of a normal to the surface S. Figure (4.8c) shows a vector existing in the whole of the volume V that is enclosed by the surface S.
Maxwell’s equations in the integral form, for =0, are obtained by taking the surface integral of Maxwell’s equation (4.4.1a):
It is assumed that a surface enclosing the magnetic field does not change with time. By using equation (4.4.12), equation (4.4.14) is changed in the following form:
(4.4.15)
where ψm is the magnetic flux. It is the Faraday law of induction that gives the induced voltage V, i.e. the emf, on a conducting loop containing the time‐varying magnetic flux, ψm:
(4.4.16)
Likewise, using Maxwell’s equation (4.4.1b) and (4.4.12) for =0, the second Maxwell’s equation is written in the integral form, giving the modified Ampere’s law:
(4.4.17)
In the above equation, Jc and Jd are conduction and displacement current densities creating the magnetic field . The above expression is generalized Ampere’s law due to Maxwell. The magnetomotive force, mmf, is obtained as follows:
where ψe is the time‐dependent electric flux. Equation (4.4.18b) is Maxwell’s induction law giving the induced mmf due to the time‐varying electric field. For the source free medium with Jc = 0, it is the complementary induction law of Faraday’s law of induction.
