147 Stirling Engine

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The construction of this stirling engine is quite simple. The acrylic container holds the working air. The air pushes the piston. The piston rotates the fly wheel. The fly wheel in turn pushes the displacer (the Styrofoam piece). The piston and the displacer are connected to the fly wheel in such a way that they are always out of phase by a quarter cycle. This automatically synchronizes the displacer to fulfill the important function of pushing the air to the hot end when it’s the air’s turn to push, and pushing the air to the cold end when it’s the fly wheel’s turn to push.

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The diagrams above show the positions of the piston and the displacer at four stages. During abc, the displacer moves the air towards the hot end so that the air is heated up. This is when we extract work done by the air. During cda, the displacer moves the air towards the cold end so that the air is cooled down. This is when work is done by the piston. Since abc occurred at higher temperature and pressure, in one complete cycle, net work done is by the air.

 

 

146 Birefringent Scotch Tape Art

Scotch tapes are birefringent. We can see them as retardation plates.

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Since the phase difference δ caused by the scotch tape’s birefringence Δn is dependent on the wavelength of the light λ, different wavelength of light are “retarded” differently. This means that each wavelength of light become polarised differently. Seen through a second (crossed) polariser, white light is turned into a color light.

Since the phase difference δ caused by the scotch tape is also dependent on the thickness of the scotch tape d, we see different colors for each thickness of scotch tape.

 

145 Stress Analysis

Many plastic, when under stress, become birefringent. These thin plastic become somewhat like retardation plates.

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Since the phase difference δ caused by the scotch tape’s birefringence Δn is dependent on the wavelength of the light λ, different wavelength of light are “retarded” differently. This means that each wavelength of light become polarised differently. Seen through a second (crossed) polariser, white light is turned into a color light. The colors thus reveal positions where the plastic is stressed.

 

144 Retardation Plates

A half-wave retarder introduces a phase difference of π radian between the o-ray and the e-ray. This has the effect of rotating the polarization angle of the EM wave by 90°.

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A quarter-wave retarder introduces a phase difference of π/2 radian between the o-ray and the e-ray. This has the effect of making a plane polarized wave into a circularly polarized wave.

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Note that the optic axis of the retarders should be diagonal to the crossed polarisers. This is so that the o-ray and e-ray are of the same amplitude. If not, we will be obtaining elliptically polarized waves instead. This explains the light level changes when retarder was rotated.

Note also that the phase difference introduced by the retarders is dependent on the wavelength of light. This explains why the color of the light changes when the quarter-wave retarder was rotated.

 

 

143 Double Refraction

A calcite crystal is a birefringent material, which is to say the crystal has not one, but two refractive indexes. This means that light traveling through the crystal can travel at one of two possible speeds, depending on the polarization of light with respect to what’s called the optic axis of the crystal.

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The result of this is that light traveling through the crystal is split into two rays: the o-ray which is polarized perpendicular to the optic axis, and the e-ray which is polarized parallel to the optic axis. Since the crystal presents two different refractive indexes to these two rays, they will be refracted to different amount, and thus emerge as two separate rays, forming the double image. Rotating a polarizer confirmed that the two images are formed by the o-ray and the e-ray which are perpendicular to each other.

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It is interesting to note that the o-ray obeys Snell’s law. So rotating the crystal does not alter the refraction of the o-ray. The refraction of the e-ray is much more complicated. In fact, rotating the crystal rotates the optic axis of the crystal as well, affecting the way the e-ray bends in the crystal. That’s why the image formed by the e-ray rotates with the crystal.

142 Disappearing Coin Trick

Make sure you did not miss the last part of the video, where the light rays were made visible. They probably offer better explanation than the text below.

This trick works better when the eye is set low.

Without the water: light from the fish undergo one air-glass-air refraction at the bottom wall, and a second air-glass-air refraction at the side wall, and arrives at the eye.

With the water: light from the fish undergo the air-glass-water refraction at the bottom wall.  Since the refractive index of glass and water are much closer (than glass and air), the ray does not quite “unbend” itself. This causes the ray to attempt the water-glass-air refraction at too large an angle. At the glass-air boundary, the ray undergoes total internal reflection, and reflects back into the tube.

Interesting notes:

  1. The total internal reflected rays can be viewed when we peep into the tube from above. It appears as the inverted fish behind the side wall.
  2. With the eye placed low (and water in the tub), what we see at the “bottom” of the glass tube (the greyish metallic surface) is actually not the bottom of the glass tube. They are probably light that have undergone total internal reflection on the bottom of the glass.

141 Reverse Arrow Trick

When filled with water, the tube basically behaves like a lens. It displays all the well known results of a convex lens. When the fish appears laterally inverted and in front of the water, it is because the fish was placed more than 1 focal length behind the water. When the fish appears magnified (but not laterally inverted) and behind the water, it is because the fish was placed less than 1 focal length behind the water.