The index of refraction of a glass rod is 1.48 at T=20.0^∘ C and varies linearly with temperatur

The index of refraction of a glass rod is 1.48 at T=20.0^∘ C...

The index of refraction of a glass rod is 1.48 at $T=20.0^{\circ} \mathrm{C}$ and varies linearly with temperature, with a coefficient of $2.50 \times 10^{-5} / \mathrm{C}^{\circ} .$ The coefficient of linear expansion of the glass is $5.00 \times 10^{-6} / \mathrm{C}^{\circ} .$ At $20.0^{\circ} \mathrm{C}$ the length of the rod is $3.00 \mathrm{~cm}$ A Michelson interferometer has this glass rod in one arm, and the rod is being heated so that its temperature increases at a rate of $5.00 \mathrm{C}^{\circ} / \mathrm{min} .$ The light source has wavelength $\lambda=589 \mathrm{nm},$ and the rod initially is at $T=20.0^{\circ} \mathrm{C}$. How many fringes cross the field of view each minute?

Key Concepts

Optical Path Length

Optical path length is the product of the physical length of a medium and its index of refraction. Changes in either the medium's length or its refractive index alter the optical path length, which is a critical factor in determining the phase of light in interferometric setups.

Interference and Fringe Shifts in Interferometry

In interferometry, such as in a Michelson interferometer, variations in optical path length lead to shifts in interference fringes. The number of fringes that cross the field of view is directly related to the change in the optical path length relative to the wavelength of the light, allowing precise measurements of small changes in physical or optical properties.

Coefficient of Linear Expansion

This coefficient quantifies how the physical dimensions of a material change with temperature. It is a measure of the fractional change in length per degree change in temperature, leading to modifications in the optical path length when the material expands or contracts.

Temperature Dependence of Optical Properties

Many materials have optical properties, such as the index of refraction, that change with temperature. A linear variation with temperature means that even modest temperature changes can cause measurable shifts in how light interacts with the material, which in turn affects experimental observations like interference patterns.

Index of Refraction

This is a fundamental property of materials that describes how light propagates through a medium. It determines the speed of light in the material relative to its speed in vacuum and affects how light bends or refracts when entering or exiting the medium.

Transcript

00:02 Okay, so here there are coefficients of rear expansion and the coefficient at which the index of refraction changes are given the index of refraction of raw litemite as tach and g is given that is 1 .48 now a 20 degrees the length of our light red does as why not the initial length is 3 centimeters so this is equal to 0 .03 meters right okay and a michael's an interferometer is used and the roar is being hit as its temperature increases okay so the rate of increase temperature is 5 celsius degrees per minute on the source of light has well -evident 589 nanometers and which is is equal to 589 into the power minus 10 meters right sorry minus 9 meters so okay so how many fringes cross the field of view each minute so we were used to crossing the fringes by actually moving the mirror right but now and the movement of mirror or the movement of glass door as in this question is due to hitting right so here by two different ways there there is the changes in fringes by two different ways and one is due to expansion of rod right so it it replaces some amount of air so one is due to expansion of roar and the change in fringes we know from the formula in the book the changes in fringes am equals to two times the moment of mirror 2 y times lambda right so in this case the change in mirror in the fringes number of fringes is equal to two times due to the expansion of the row so dev y right upon lambda g on lambda g is what lambda upon ita g or n g where i have used this relation here lambda g equal to lambda upon n g where lambda g is the wavelength applied inside the glass now minus two times del y and now in here on lambda now what is the change in then they'll why so let me keep that common here two times n g minus n and then why what is del y del y is the initial length times alpha times changing ten temperature del theta and it is given that del theta the change in temperature is 5 centigrade per minute so let me write here 5 degrees and degree we are all solving in period of 1 minutes so this upon alpha now you know the value of ng you know the value of n a right here refractive index of air equals to 1 we know why not equal to 0 .03 we know alpha alpha is given in the question that is equal to 5 into 10 to the power minus 6 per degree celsius right and we know del theta we know lambda just plug in all the values in here so this gives me del m equals to 1 .22 okay and the second reason for the fringes for the movement of fringes is due to the change in detractive of rod due to change in refractive index of rod and this is due to del m equals to two times the change in refractive index right into why not the initial length upon lambda you have to note that since the effect is very small we can consider these two separately and not at the same time okay so this is equal to 2 into now the change in the interactive index of rod is given the coefficient is 2 .5 into 10 to bar minus 5 per degree celsius so since the change is 5 so this should be 5 into 2 .5 into 10 to the bar minus 5 right because this is per degree celsius and this is the degrees that is changed now into y -not is again 0 .0...

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