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Applications of the Laplace Transform


Theoretical Approach

In figure 1, mechanical transverse waves sent along a rope pass through a slot. Under these circumstances, waves can be sent along the rope only if the waves are generated parallel to the direction of the slot. Each slot permits only

those waves with the proper orientation to pass through. The waves are said to be polarized in a particular plane or plane polarized. Vertically polarized waves cannot pass through an horizontal polarizer ( Fig.1.)

Before the introduction of the electromagnetic theory, light was assumed to be a longitudinal wave disturbance. Electromagnetic theory predicts that light is a transverse wave. From interference and diffraction experiments, we can infer nothing concerning the transverse nature of light. Only polarization theory and experiments can support the theoretical electromagnetic prediction that light waves are transverse.

Fresnel observed that a beam of light falling on a calcite crystal (CaCO3) was separated into two beams that were incapable of producing interference fringes (Fig.2).

Young suggested that this could be explained by assuming that light consisted of transverse waves that were separated into component waves having oscillating planes at right angles to each other. He called this a plane polarization effect, and the phenomenon is called double refraction. The ordinary wave (o-wave) travels in the crystal with the same speed vo in all directions, the crystal having a single index of refraction no. The extraordinary wave (e-wave) travels in the crystal with a speed that is greater than vo. The index of refraction (c/ve) has a smaller value ne. Several doubly refracting crystals, other than calcite, are:

ice (H2O); quartz (SiO2); wurzite (ZnS); dolomite (CaO.MgO.2Co2); siderite (Fe.CO2)

Fig. 3 shows an electromagnetic plane-polarized wave.

The vibrations of the E vector are parallel to each other for all points in the wave. At any such a point, E and the direction of propagation form a plane called plane of vibration. In a plane polarized wave, all such planes are parallel.

The light propagated in a given direction consists of independent wavetrains whose planes of vibration are randomly oriented about the direction of propagation. Such light, though still transverse, is unpolarized (See Fig.4.).

Fig. 1 Polarized mechanical waves

Fig. 2 Double refraction of light accross a calcite crystal

Fig.3 A plane-polarized electromagnetic wave

Fig.4 Observing polarised light

Certain crystaline substances transmit light in one plane of polarization and absorb light in other polarization planes. Tourmaline is such a material. This property of crystals in which one polarized component of incident light is absorbed and the other is transmitted is called dichroism. Dichroic crystals of quinine iodosulphate transmit plane polarized light very efficiently but the crystals are too small for practical use. There is a method of embedding these crystals in cellulose films so that the dichroic properties of the crystal are retained. This polarizing film is known commercially as Polaroid.

Ordinary light incident obliquely on the surface of a glass plate is partly reflected and partly refracted. Both the transmitted and the reflected beams are partly polarized (Fig. 5.).

Fig. 6 Transmission and refraction of light trough a calcite crystal

Fig.5 Partly polarized reflected and refracted light beams

The component of the incident light lying in the plane parallel to the surface of the glass is largely reflected. The component in the plane perpendicular to the surface is largely refracted. A particular angle af incidence at which polarization of the reflected light is complete, known as the polarization angle, can be found experimentally.

Calcite crystals (Iceland spar) are sometimes polished, cut through and cemented back together in such a way that one of the polarized beams is totally reflected at the cemented face. Such a crystal, known as a Nicol prism, can be used to produce aa beam of completely polarized light (Fig.6)

While rotating the Nicol about the incident ray, the plane of vibration is also rotated.

If a bunch of natural light rays passes through a polarizer, prior to pass through an analizer, the intensity of the emerged ray is given by the law of Malus:

being the angle between the planes of vibration of light before and after the analizer, Io - the intensity of light entering the analizer.

Experimental Procedure

Fig.7 The device used for polarisation

The scheme of the device used is presented in fig. 7. Light from the source S reaches the polarizer N. The Nicol can be rotated about a vertical axis and the resulted light can be analized by the analizer A ( a plane mirror mounted at

45 o).

The intensity of the reflected light is measured by means of a photosensitive cell which generates electrical currents proportional to the intensity of the luminous flux incidental on the cathode.The photocell is placed in the focus of a lens L.

The Nicol tube has to be rotated about a vertical axis, 30o by 30o up to 360o and the intensity of the resulted current will be measured for each position of the Nicol. Results will be recorded in the data table.

Data table


One plots and in polar coordinates.

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