Simply Lecture

The Maxwell fourth equation is called "the modified Ampere's circuital law". Statement: It states that the line integral of the magnetic field H around any closed part or circuit is equal to the current enclosed by the path. Differential form (without modification): That is, $$\int H.dL = I$$ Let the current be distributed through the current with current density J, then: $$I = \int J. ds$$ This implies that: $$\int H.dL = \int J.ds$$ .........(9) Applying Stokes theorem to the LHS of equ(9) to change line integral to surface integral we have: $$\int_{s} (\nabla X H).ds = \int_{s} J.ds$$ Since, two surface integrals are equal only if their integrands are equal. Thus, $$\nabla X H =J$$ .........(10) Equ(10) is the differential form of Maxwell fourth equation (without modification) Take divergence of equ(10) $$\nabla.(\nabla X H) = \nabla.J$$ Since, the divergence of the curl of a vector is zero.  Therefore, $$\nabla.(\nabla X H) = 0$$ It means that $...

Also called the Faraday law of "electromagnetic induction". The Maxwell third equation has two statements. Statement I:  It states that whenever a magnetic flux link with a circuit changes, then induced electromotive force (emf)  is set up in the circuit. Statement II: The magnitude of induced emf is equal to the rate of magnetic flux linked with the circuit. Integral form: Therefore; $$Induced-emf = - \frac {d\psi_m}{dt}$$ where, $$\psi_m = \int B.ds$$   ......(5) The negative sign is because of Lentz law,  which states that the induced emf set up a current in such a direction that the magnetic effect produced by it opposes the cause producing it. Also, the definition of emf states that the emf is the closed line integral of the non conservative electric field generated by the battery. That is: $$emf = \int E.dL$$ ........(6) Comparing equ(5) and equ(6) we have: $$\int E.dL = -\int_{s} \frac{dB.ds}{dt}$$ ...(7) Differential form Applying Sto...

The Maxwell second law is also called "Gauss law of magnetism". Statement: It states that the total magnetic flux $$\psi_m$$ emerging through a closed surface is zero. Integral form:  $$\psi_m = \int B.ds = 0$$ .......(3) Equ (3)  is the integral form of the Maxwell equation. This equation also proves that the magnetic monopole does not exist. Differential form: Apply Gauss divergence theorem to equ (3). That is: $$\int_{s} B.ds = \int_{v} (\nabla.B) dV$$ Since: $$\int B.ds =0$$ Thus: $$\nabla.B = 0$$ .......(4) Equ(4) is the differential form of the Maxwell second equation.

The Maxwell first equation in electrostatics is called the Gauss law in electrostatics. Statement:  It states that the total electric flux \(\psi_E\) passing through a closed hypothetical surface is equal to \(\frac{1}{\epsilon_0}\) enclosed by the surface. Integral Form: $$\phi_E = \int E.ds = \frac{q}{\epsilon_0}$$ $$\int D.ds = q$$ where, $$D = \epsilon_0 E = displacement-vector$$ Let the change be distributed over a volume v and \(\rho\) be the volume charge density. Hence, $$q = \int \rho dv$$ Therefore; $$\int D.ds = \int_{v} \rho dv$$ .........(1) Equ(1) is the integral form of Maxwell first law Differential form: Apply Gauss divergence theorem to the L.H.S of equ(1) from surface integral to volume integral. $$\int D.ds = \int (\nabla.D)dv$$ Substituting this equation to equ(1) $$\int(\nabla.D)dv = \int_{v} \rho dv$$ As two volume integrals are equal only if their integrands are equal. Thus; $$\nabla.D = \rho v$$ ............(2) Equ(2) is the dif...