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Maxwell's Equations for Time-Varying Fields

In addition to his contribution of displacement current, Maxwell brought together the four basic laws governing electric and magnetic fields into one set of four equations which, as a set, completely describe the behaviour of any electromagnetic field. All of the vector field, flux, current and charge terms in Maxwell’s equations are, in general, functions of both time and space [e.g., E (x, y, z, t)]. The form of these quantities is referred to as the instantaneous form (we can describe the fields at any point in time and space). The instantaneous form of Maxwell’s equations may be used to analyse electromagnetic fields with any arbitrary time variation.
 
Maxwell’s equations
 
Sl. No. Differential Form Integral Form Name
1. Description: 19325.png Description: 17706.png Gauss’ law of electrostatics
2. Description: 19827.png Description: 18466.png
Gauss’ law of magnetostatics
(non-existence of magnetic monopole)
3. Description: 20090.png Description: 17565.png Faraday’s law of electromagnetic induction
4. Description: 17398.png Description: 18995.png Modified Ampere’s circuital law

Time Harmonic Fields

Time harmonic fields are those fields that vary sinusoidally with time. They are easily expressed in phasors.

Maxwell’s Equations for Time-Harmonic Fields

To transform the instantaneous Maxwell’s equations into time-harmonic forms, we replace all sources and field quantities by their phasor equivalents and replace all time derivatives of quantities with jω times the phasor equivalent. Thus, the Maxwell’s equations for time harmonic fields are given in the table.
 
Maxwell’s equations for time harmonic fields
 
Sl. No. Differential Form Integral Form Name
1. Description: 17487.png Description: 19205.png Gauss’ law of Electrostatics
2. Description: 17819.png Description: 19592.png
Gauss’ law of Magnetostatics
(non-existence of magnetic monopole)
3. Description: 16570.png Description: 16463.png Faraday’s law of electromagnetic induction
4. Description: 17383.png Description: 18123.png Modified Ampere’s circuital law




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