The Calculus Reveals Special Properties of Light
Richard D. Sauerheber
Department of Chemistry, University of California, San Diego, La Jolla, CA 92037 U.S.A.
Department of Tutorial Services, Palomar Community College, San Marcos, CA 92069 U.S.A.
Abstract. Sequential derivatives of the sine wave indicate the mechanism by which electric and magnetic fields in light beams self induce where the acceleration of the acceleration of each cycles in rhythmic continuity. Light is described as perpendicular electric-magnetic vectors that oscillate perpendicular to forward propagation while also undergoing circular rotation. The difference between intrinsic speed c and propagation velocity, verses relative speed and relative velocity is presented. The functions of light in the natural world may be better understood as a result of this analysis.
Introduction. Applications of the Calculus to the physical world have been extremely diverse and productive. Previous articles presented various Calculus features of a light wave (Sauerheber, 2010, 2011). Here, the properties of the sine function and its Calculus are described that delineate the intrinsic behavior of light. Light waves themselves may be both amplitude and frequency modulated independently, where electronic acceleration and deceleration events in stars between differing states of energy produce light of variable amplitude and frequency. However intrinsic propagation speed for light is fixed at c, as determined by Maxwell (Giancoli, 2009) in the 19th century from his famous equations of state for all massless electromagnetic radiation. Thus, the wavelengths (l) of light resulting from such widely varying electronic transitions must follow the relation l = c/f, where f is the frequency of the light, directly proportional to its intrinsic energy content hf (h being Planck‘s constant).
It is important to emphasize the power of the Calculus in describing aspects of the physical world reliably. After Maxwell’s major triumph, Albert Michelson in the San Gabriel Mountains, California performed difficult experiments with a rotating slotted mirror for light beams traveling 70 km round-trip that confirmed to seven digit precision the Maxwell value for light speed, as c = 1/(em)1/2, by direct measurement (Giancoli, 2009, Beiser, 1963, Sauerheber, 2007). e and m are the electrical permittivity and magnetic permeability for the given medium. Maxwell’s constitutive and field equations prove that light must travel as sinusoidal waves of alternating self-induced perpendicular electric and magnetic fields, and we now know light waves travel in vacuum in the complete absence of a material substance. Michelson’s data confirm that speed c reflects the propagation speed along the wavelength axis.
The Calculus of Light. A deep understanding of the physics of a light wave is possible by considering the properties of the sine function, that is descriptive for light as shown in Figure 1, and its multiple derivatives. From Newton we know that velocity and acceleration functions are respective sequential derivatives of a position function. From Maxwell we know that the sin(x) graph delineates the variations in electric and magnetic field intensities as a function of position along a light wave, where intensity is 0 at wave position 0 and maximal at p/2. The derivative of the sine function reports the rate of change of the field intensity along the wave, where d[sin(x)]/dx = cos(x). Thus the cos(x) graph indicates the velocity with which the field intensity changes and is maximum at 0 but slows to 0 at p/2. Therefore a negative acceleration slows the velocity, from maximum to zero, and this is borne out in the second derivative, where d2[sin(x)]dx2 = -sin(x). This upside down sin(x) curve indicates the acceleration is negative between 0 and p /2, zero at 0, and maximum negative at p/2. Thus the electric and magnetic field intensity at position 0 has maximum velocity (with zero acceleration at this position) that is responsible for the increasing value of intensity that proceeds along the wave direction.
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