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Question
Two particles X and Y having equal charge, after being accelerated through the same potential difference enter a region of uniform magnetic field and describe circular paths of radii R1 and R2 respectively. The ratio of the mass of X to that of Y is ______.
Options
(R1/R2)1/2
R1/R2
(R1/R2)2
R1R2.
Solution
Two particles X and Y having equal charge, after being accelerated through the same potential difference enter a region of uniform magnetic field and describe circular paths of radii R1 and R2 respectively. The ratio of the mass of X to that of Y is (R1/R2)2.
Explanation:
`"R"_1^2/"R"_2^2`
Particles X and Y of respective masses m1 and m2 are carrying charge q describing circular paths with respective radii R1 and R2 such that
`"R"_1 = ("m"_1"v"_1)/"qB"`
`"R"_2 = ("m"_2"v"_2)/"qB"`
Since both the particles are accelerated through the same potential difference, both will have the same kinetic energy.
`therefore 1/2 "m"_1"v"_1^2 = 1/2 "m"_2"v"_2^2`
`because "R"_1 = ("m"_1"v"_1)/"qB" => "v"_1 = ("R"_1"qB")/"m"_1`
And
`because "R"_2 = ("m"_2"v"_2)/"qB" => "v"_2 = ("R"_2"qB")/"m"_2`
`therefore "m"_1 (("R"_1 "qB")/"m"_1)^2 = "m"_2 (("R"_1 "qB")/"m"_1)^2`
`=> "m"_1/"m"_2 = "R"_1^2/"R"_2^2`
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