Question

Derive maxwell's divergence equation from maxwell's curl equation.​

Answer 1

Answer:

Maxwell First Equation

Maxwell first equation is based on the Gauss law of electrostatic which states that “when a closed surface integral of electric flux density is always equal to charge enclosed over that surface”

Mathematically Gauss law can be expressed as,

Over a closed surface the product of electric flux density vector and surface integral is equal to the charge enclosed.

∯[latex]\vec{D}.d\bar{s}=Q_{enclosed}[/latex]      ——(1)

Any closed system will have multiple surfaces but a single volume. Thus, the above surface integral can be converted into a volume integral by taking the divergence of the same vector. Thus, mathematically it is-

∯ [latex]\bar{D}.d\bar{s}=\iiint \bigtriangledown .\vec{D}d\vec{v}[/latex] —-(2)

Thus, combining (1) and (2) we get-

[latex]\iiint \bigtriangledown .\vec{D}d\vec{v}=Q_{enclosed}[/latex] —–(3)

Charges in a closed surface will be distributed over its volume. Thus, the volume charge density can be defined as –

[latex]\rho v=\frac{dQ}{dv}[/latex]  measured using C/m3

On rearranging we get-

[latex]dQ=\rho vdv[/latex]

On integrating the above equation we get-

[latex]Q=\iiint \rho vdv[/latex] ——-(4)

The charge enclosed within a closed surface is given by volume charge density over that volume.

Substituting (4) in (3) we get-

[latex]\iiint\bigtriangledown .D dv =\iiint \rho vdv[/latex]

Canceling the volume integral on both the sides, we arrive at Maxwell’s First Equation-

[latex]\Rightarrow \bigtriangledown .D dv = \rho v[/latex]

Explanation:

Answer 2

Answer:

.Maxwell's 3rd equation is derived from Faraday's laws of Electromagnetic Induction. It states that “Whenever there are n-turns of conducting coil in a closed path which is placed in a time-varying magnetic field, an alternating electromotive force gets induced in each and every coil.” This is given by Lenz's law.

Hope it helps u !!