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A charged particle (q,m) enters uniform...

A charged particle `(q,m)` enters uniform magnetic field B at angle `alpha` shown in figure with speed `v_0`. Find

a. The angle `beta` at which it leves the magnetic field
b. time spent by the particle in magnetic field and
c. the distance `C`.

Text Solution

Verified by Experts

a. Here, velocity of the particle is in the plane of paper while the magnetic field is perpendicular to the paper inwards, i.e. angle between v and B is `90^@`. So the path is a circle. The radius of the circle is `r=(mv_0)/(Bq)`

`O` is the centre of the circle. In `/_\AOC`
`/_OCD=/_OAD`
`or 90^@-beta=90^@-alpha`
`:. beta=alpha`
b. `/_COD=/_DOA=alpha` (as `/_OCD=/_OAD=90^@-alpha)`
`:. /_AOC=2alpha`
or length `APC=r(2alpha)=(2mv_0)/(Bq) .alpha`
`:. t_(APC)=(APC)/v_0=(2malpha)/(Bq)`
Alternate method
`t_(APC)=(T/(2pi))(2alpha)=(alpha/pi).T`
`=(alpha/pi)((2pim)/(Bq))=(2alpham)/(Bq)`
c. Distance `AC=2 (AD)=2(rsinalpha)`
`=(2mv_0)/(Bq)sinalpha`
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