The index of refraction of a glass plate is $$1.48$$ at θ$$1=300C$$ and varies linearly with temperature with a coefficient of $$2.5$$×$$10$$−$$5$$ $$0C$$−$$1$$ . The coefficient of linear expansion of the glass is $$5$$×$$10$$−$$5$$ $$0C$$−$$1$$ at $$300C$$ , the length of the glass plate is $$3cm$$. this plate is placed in front of one of the sits in Young’s $$double-slit$$ experiment. If the plate is being heated so that its temperature increases at a rate of $$5$$ $$0C$$−$$1min$$ , the light source has wavelength λ$$=589nm$$ and the glass plate initially is at θ$$=300C$$ . The number of fringes that shift on the screen in each minute is nearly (use$$ $$approximation)

Question

The index of refraction of a glass plate is $$1.48$$ at θ$$1=300C$$ and varies linearly with temperature with a coefficient of $$2.5$$×$$10$$−$$5$$ $$0C$$−$$1$$ . The coefficient of linear expansion of the glass is $$5$$×$$10$$−$$5$$ $$0C$$−$$1$$ at $$300C$$ , the length of the glass plate is $$3cm$$. this plate is placed in front of one of the sits in Young’s $$double-slit$$ experiment. If the plate is being heated so that its temperature increases at a rate of $$5$$ $$0C$$−$$1min$$ , the light source has wavelength λ$$=589nm$$ and the glass plate initially is at θ$$=300C$$ . The number of fringes that shift on the screen in each minute is nearly (use$$ $$approximation)

Infinity Learn · Accepted Answer

μ$$0=refractive$$ index of glass plateat $$30$$°Cμ$$=final$$ refractive index of glass plate after $$heatingt0=initial$$ thickness of glasss plate at $$30$$°$$Ct=final$$ thickness of glass plate after heatingPath difference $$=$$μt−μ$$0t0=n$$λ$$0$$ Where n is the number of fringes that shift on the screen⇒μ$$0(1+$$α$$1$$θ$$)t0(1+$$α$$2$$θ$$)$$−μ$$0t0$$λ$$0=n$$μ$$0t0($$α$$1+$$α$$2)$$θλ$$0=nGiven$$, μ$$0=1.48$$,$$t0=3$$×$$10$$−$$2m$$α$$1=2.5$$×$$10$$−$$5$$  $$0C$$−$$1$$α$$2=0.5$$×$$10$$−$$8$$ $$0C$$−$$1$$,θ$$=50min$$−$$1$$λ$$0=589nm$$∴$$n=1.48$$×$$3$$×$$10$$−$$2(3$$×$$10$$−$$5)$$×$$5589$$×$$10$$−$$9=11$$

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