# 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 Maxwellâ€™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 Maxwellâ€™s equations may be used to analyse electromagnetic fields with any arbitrary time variation.

Maxwellâ€™s equations

 Sl. No. Differential Form Integral Form Name 1. Gaussâ€™ law of electrostatics 2. Gaussâ€™ law of magnetostatics (non-existence of magnetic monopole) 3. Faradayâ€™s law of electromagnetic induction 4. Modified Ampereâ€™s circuital law

# Time Harmonic Fields

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

# Maxwellâ€™s Equations for Time-Harmonic Fields

To transform the instantaneous Maxwellâ€™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 jÏ‰ times the phasor equivalent. Thus, the Maxwellâ€™s equations for time harmonic fields are given in the table.

Maxwellâ€™s equations for time harmonic fields

 Sl. No. Differential Form Integral Form Name 1. Gaussâ€™ law of Electrostatics 2. Gaussâ€™ law of Magnetostatics (non-existence of magnetic monopole) 3. Faradayâ€™s law of electromagnetic induction 4. Modified Ampereâ€™s circuital law