Skip to main content

Maxwell first 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 differential form of Maxwell first equation.
Labels:

Comments

Post a Comment

Popular posts from this blog

Maxwell fourth equation

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 $...
Post a Comment
Read more

Time dependent Schrodinger equation

Recall from the time independent Schrodinger equation: $$-\frac{\hbar^{2}}{2m} \nabla^{2} \psi(r) + V(r) \psi(r) = E \psi(r)$$ ........(1) Multiply both sides of equ (1) by \(\phi(t)\), we have: $$-\frac{\hbar^{2}}{2m} \nabla^{2} \psi(r) \phi(t) + V(r) \psi(r) \phi(t) = E \psi(r) \phi(t)$$ ..........(2) Recall: $$\psi(r) \phi(t) = \Psi(r,t)$$ .............(3) Substitute equ (3) into equ (2): $$-\frac{\hbar^{2}}{2m} \nabla^{2} \Psi(r,t) + V(r) \Psi(r,t) = E \Psi(r,t)$$ .........(4) Recall that the well known Planck-Einstein relation is given by: $$E = hv$$ ...........(5) And also, the reduced Planck's constant is given by: $$\hbar = \frac{h}{2\pi}$$ ................(6) Also recall that time dependence is given by: $$\phi(t) = C_+ exp(-i2\pi vt) + C_- exp(i2\pi vt)$$ ..........(7) Substituting equ (6) into (5), we have: $$V = \frac{E}{\hbar 2 \pi}$$ ..........(8) Substitute equ (8) into equ (7) $$\phi(t) = C_+ exp(-i2 \pi \frac{E}{\hbar 2 \pi} t)  + C_- exp(i2 \pi \f...
Post a Comment
Read more

Angel flight

Angel flight is a term used by group of people whose members are provided with free air transportation,  because they are in need with free medical treatment far from home. The transportation of passengers are done by volunteer pilots using their own general aviation aircraft. History of angel flight  The first two organisation to be termed "angel flight" was founded in the year 1983. The first organisation was formed in Santa Monica, California known as the "Angel flight of California (presently Angel Flight West)". The second was formed in Atlanta, Georgia and was called "Angel Flight Soars" Accidents and incidents On 15th August, 2011 a Piper PA-28 Cherokee conducting an angel flight crashed in rural Victoria, Australia. On May 24, 2013 an angel flight crashed into a pond in Ephratah, New York. On June 28, 2017 a TBIO Tobago serving an angel flight crashed into a terrain near MT Gambier heading to Adelaide. Pilots  The pilots of angel flig...
Post a Comment
Read more