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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.
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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