There are all sorts of resonances around us, in the world, in our culture, and in our technology. A tidal resonance causes the 55 foot tides in the Bay of Fundy. Mechanical and acoustical resonances and their control are at the center of practically every musical instrument that ever existed. Even our voices and speech are based on controlling the resonances in our throat and mouth. Technology is also a heavy user of resonance. All clocks, radios, televisions, and gps navigating systems use electronic resonators at their very core. Doctors use magnetic resonance imaging or MRI to sense the resonances in atomic nuclei to map the insides of their patients. In spite of the great diversity of resonators, they all share many common properties. In this blog, we will delve into their various aspects. It is hoped that this will serve both the students and professionals who would like to understand more about resonators. I hope all will enjoy the animations.

Origins of Newton's laws of motion

History of mechanical clocks with animations
Understanding a mechanical clock with animations
includes pendulum, balance wheel, and quartz clocks

Sunday, July 17, 2011

Transforming the second, third and fourth Maxwell equation

17. Transforming the second, third and fourth Maxwell equations

Transforming the second Maxwell equation

We start transforming the second Maxwell equation. This is a vector equation:

Because of its complexity, we now finish the transformation component-by-component:

Note that 1/γ2 = 1 − β2 where β = V/c  (see the definition of  γ ).

Transformation of the third Maxwell equation

The third Maxwell equation transforms similarly to the first one above:

Transformation of the fourth Maxwell equation

The fourth Maxwell equation is a vector equation and will transform in a similar manner to the second Maxwell equation above. We start by transforming the left side of the fourth Maxwell equation:

Now we transform this component-by-component:

The right side of the fourth Maxwell equation is transformed as:

where we have used the relation:   c2 = 1/(ε0μ0 ) which can be solved for μ0, i.e.: μ0 = 1/(c2ε0) .

Combining the above equations, we get: