Maxwell's Equations and their Physical Significances

By studying the physical properties of electric (E-field) and magnetic (B-field) fields we have been able to describe these properties by four, relatively simple, equations known as Maxwell’s equations.

These four fundamental equations of electromagnetism can be expressed in both an integral and differential form as tabulated below:

Coulomb’s and Gauss’s Laws

Equation (1) results from Coulomb’s and Gauss’s Laws and states that free charges act as sources or sinks of D. It suggests that the total electric flux density or total electric displacement through the surface enclosing a volume v is equal to the total charge within the volume.

Equation (2) arises from the application of Gauss’s law to magnetic fields and the non-existence of magnetic monopoles. There are no sources or sinks of B. This equation suggests that the net magnetic flux emerging through any closed surface is zero.

Equation (3) describes Faraday’s Law of electromagnetic induction and states that an electromotance is produced in a circuit when the magnetic flux through the circuit changes. It suggests that the electromagnetic force around a closed path is equal to the time derivative of the magnetic flux density through any surface bounded by the path.

Equation (4) describes Ampere’s Circuital Law (which is derived from the Biot-Savart Law) and states that the electromotive force around a closed path is equal to the conduction current J = σE plus the time derivative of the electric flux density through any surface bounded by the path.

Maxwell’s equations

In vacuum/free space ρ_v = 0, J = 0, ε = ε_o and μ = μ_o. Therefore, in vacuum the Maxwell’s equations take the following forms:

Maxwell's Equations in Frequency Domain

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Assuming the fields are varying harmonically with time as e^jωt, the Maxwell’s equations are given by

Maxwell’s equations

 

Maxwell's Equations in Phasor Form

In case the field quantities are sinusoidally time varying; the electric field E can be expressed as

E(x, y, z, t) = E_x (x, y, z, t)a_x +E_y (x, y , z, t) a_y + E_z (x, y, z, t) a_z

Where E_x = E_xm cos(ωt+θ_x), E_y = E_ym cos(ωt+θ_y), E_z = E_zm cos (ωt+θ_z). Here the magnitudes E_xm, E_ym, E_zm, and the phase angles θ_x, θ_y, θ_z, are independent of time but may depend on spatial coordinates, e. g., E_xm (x, y, z), θ_x(x, y, z). Now E_x can be expressed as

Maxwell's Equations

That is, the first derivative of a sinusoidal varying field is jω times the field. Therefore, the Maxwell’s equations in phasor form can be expressed as:

Maxwell's Equations in Phasor Form