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A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carry an intensity I , find the force exerted by the beam on the sphere?
Verified Answer
A perfectly reflecting solid sphere of radius r is kept in the path of...
Explanation :
Let O be centre of sphere and OZ be line opposite to incident beam . Consider a radius OP of sphere making an angle θ with Oz to get aciroll on sphere. Charge θtoθ+dθ and rotate radius about OZ to get another circle on sphere . The part of sphere between these circles is a ring of area 2πr2sinθdθ. Consider a small part △A of this right at P . Energy of light falling on this part in time △t is
△u=I△t(△Acosθ)
The momentum of this light falling on △A is △u along QP. The light e is reflected by sphere along PR .Therefore in momentum is
△P=2△uCcosθ=2CI△t(△Acos2θ) (direction along OP−→−)
The force on △A due to light falling on it , is
△P△=2CI△Acos2θ (directed along PO−→−)
The resultant force on ring as well as an sphere is along ZO by force on △A along ZO.
△P△tcosθ=2CI△Acos3θ (along ZO−→−)
The force acting on ring is
dF=2CI(2πr2sinθdθ)cos3θ
The force on entire sphere is
F=∫π204πr2ICcos3θsinθdθ
=−∫π204πr2ICcos3θd(cosθ)
=−∫π204πr2IC[cos4θ4]π20
=πr2IC
Vote that integration is done only for hemisphere that faces the incident beam.
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A perfectly reflecting solid sphere of radius r is kept in the path of...
Force exerted by a beam of light on a perfectly reflecting solid sphere
The force exerted by a beam of light on a solid sphere can be calculated using the principle of radiation pressure. When a beam of light strikes a surface, it exerts a force on it due to the transfer of momentum. This force can be calculated using the formula:
Force = Power / Speed of light
To determine the force exerted by the beam on the sphere, we need to calculate the power of the beam incident on the sphere.
Power incident on the sphere
The power incident on the sphere can be calculated by considering the intensity of the beam and the area over which it is incident. Since the beam has a large aperture, we can assume that the entire beam is incident on the sphere.
The intensity of the beam is given as I, which represents the power per unit area. Therefore, the power incident on the sphere can be calculated by multiplying the intensity by the area of the sphere's cross-section.
Area of the sphere's cross-section
The cross-section of a sphere is a circle. The area of a circle is given by the formula:
Area = π * r^2
where r is the radius of the sphere.
Therefore, the area of the sphere's cross-section is given by:
Area = π * r^2
Calculating the power incident on the sphere
Now we can calculate the power incident on the sphere by multiplying the intensity of the beam by the area of the sphere's cross-section.
Power = Intensity * Area
Power = I * (π * r^2)
Calculating the force exerted by the beam
Finally, we can calculate the force exerted by the beam on the sphere by dividing the power incident on the sphere by the speed of light.
Force = Power / Speed of light
Force = (I * π * r^2) / c
where c is the speed of light.
Conclusion
In conclusion, the force exerted by a beam of light on a perfectly reflecting solid sphere can be calculated using the formula: Force = (I * π * r^2) / c, where I is the intensity of the beam, r is the radius of the sphere, and c is the speed of light. The force exerted is directly proportional to the intensity of the beam and the area of the sphere's cross-section.
The force exerted by a beam of light on a solid sphere can be calculated using the principle of radiation pressure. When a beam of light strikes a surface, it exerts a force on it due to the transfer of momentum. This force can be calculated using the formula:
Force = Power / Speed of light
To determine the force exerted by the beam on the sphere, we need to calculate the power of the beam incident on the sphere.
Power incident on the sphere
The power incident on the sphere can be calculated by considering the intensity of the beam and the area over which it is incident. Since the beam has a large aperture, we can assume that the entire beam is incident on the sphere.
The intensity of the beam is given as I, which represents the power per unit area. Therefore, the power incident on the sphere can be calculated by multiplying the intensity by the area of the sphere's cross-section.
Area of the sphere's cross-section
The cross-section of a sphere is a circle. The area of a circle is given by the formula:
Area = π * r^2
where r is the radius of the sphere.
Therefore, the area of the sphere's cross-section is given by:
Area = π * r^2
Calculating the power incident on the sphere
Now we can calculate the power incident on the sphere by multiplying the intensity of the beam by the area of the sphere's cross-section.
Power = Intensity * Area
Power = I * (π * r^2)
Calculating the force exerted by the beam
Finally, we can calculate the force exerted by the beam on the sphere by dividing the power incident on the sphere by the speed of light.
Force = Power / Speed of light
Force = (I * π * r^2) / c
where c is the speed of light.
Conclusion
In conclusion, the force exerted by a beam of light on a perfectly reflecting solid sphere can be calculated using the formula: Force = (I * π * r^2) / c, where I is the intensity of the beam, r is the radius of the sphere, and c is the speed of light. The force exerted is directly proportional to the intensity of the beam and the area of the sphere's cross-section.
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A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carry an intensity I , find the force exerted by the beam on the sphere?
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A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carry an intensity I , find the force exerted by the beam on the sphere? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carry an intensity I , find the force exerted by the beam on the sphere? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carry an intensity I , find the force exerted by the beam on the sphere?.
A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carry an intensity I , find the force exerted by the beam on the sphere? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carry an intensity I , find the force exerted by the beam on the sphere? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for A perfectly reflecting solid sphere of radius r is kept in the path of a parallel beam of light of large aperture. If the beam carry an intensity I , find the force exerted by the beam on the sphere?.
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