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A sphere of radius R is exposed to a parallel beam of radiation of intensity I as shown in figure. Choose the correct option (s) of the following. |
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Answer» If the surface of the sphere is completely reflecting, radiation force in the sphere is `(2IpiR^(2))/(c )`  CONSIDER a circular strip of radius `R sin theta` and of width `Rd theta`. AMOUNT of energy falling on the strip per sec `=IdAcos theta` Change in momentum due to reflection `dp=(2I)/(c )(DACOS theta)cos theta` Force on the strip `dF=(2I)/(c )dAcos^(2)theta=(2I)/(c )xx2piRsin thetaxxRd thetaxxcos^(2)theta` Net force on the sphere `F=int_(0)^((pi)/(2))dF costheta=int_(0)^((pi)/(2))(4piIR^(2))/(c )cos^(3)thetasinthetad theta` `=-(4piIR^(2))/(c )int_(0)^((pi)/(2))cos^(3)theta(-sinthetad theta)` `=-(4piIR^(2))/(c )[(cos^(4)theta)/(4)]_(0)^((pi)/(2))=(piIR^(2))/(c )[cos^(4)theta]_((pi)/(2))^(0)=(IpiR^(2))/(c )` When surface is completly absorbing `dF=(I(dAcostheta))/(c )` Net force `F=intdF=int_(0)^((pi)/(2))(I2piRsintheta(Rd theta)costheta)/(c )` `=(IpiR^(2))/(c )int_(0)^((pi)/(2))sin2thetad theta=(IpiR^(2))/(c )[(-cos2theta)/(2)]_(0)^((pi)/(2))` `=(IpiR^(2))/(2c )[cos2theta]_((pi)/(2))^(0)=(IpiR^(2))/(2c )xx2=(IpiR^(2))/(c )` when surface is partially reflecting with reflection coefficent `0.3` and absorbtion coefficent `0.7` net force on the sphere is `F=int_(0)^((pi)/(2))(2(0.3l)(dAcostheta)costheta)/(c )xxcostheta+int_(0)^((pi)/(2))((0.7l)(dAcostheta))/(c )` `= 0.3(IpiR^(2))/(c )+0.7(IpiR^(2))/(c )=(IpiR^(2))/(c )` Note: In all the above cases radiation force=radiation pressure (due to absorbtion)xxeffective area perpendicular to the flow of energy`=(I)/(c )xxpiR^(2)` |
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