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.

Answer 1

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`