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Question
Answer the following question.
A charged particle q is moving in the presence of a magnetic field B which is inclined to an angle 30° with the direction of the motion of the particle. Draw the trajectory followed by the particle in the presence of the field and explain how the particle describes this path.
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Solution
Lorentz force will be exerted on the particle because of the Sine component of particle velocity thereby making the particle move in a helical path.
Time period = `("Total distance")/("speed") = (2"πR")/("v"_"y") (2"πR")/("v" sin 30°) = (4"πR")/("v")`
Pitch = speed along x-axis × time period
`= "v"_x xx (2"πR")/("v"_"y")`
`= "v"_x xx (2"πR")/("v"sin30°) = "v"_x xx (2"πR" xx 2)/("v")`
= `"v" cos 30° xx (4"πR")/("v") = sqrt(3)/(2) xx 4"πR" = 2sqrt(3)"πR"`
R = `(mv sin 30°)/(q"B")`
Pitch = `(2sqrt(3)"πmv" sin 30°)/(q"B")`
= `(2sqrt(3)"πmv")/(q"B" xx2)`
= `(sqrt(3)"πmv")/(q"B")`
The vertical component of velocity will make the charged particle to move in a circular path whereas the horizontal component of velocity will provide Pitch = `(sqrt(3)"πmv")/(q"B")` Hence the motion of the particle will be helical with the Pitch= `(sqrt(3)"πmv")/(q"B")`.
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