Question

A researcher studying the properties of ions in the upper atmosphere wishes to construct an apparatus with the following charateristics: Using an electric field, a beam of ions, each having charge q, mass m , and initial velocity `vhati`, is turned through an angle of `90^@` as each ion undergoes displacement `Rhati + Rhatj`. The ions enter a chamber as shown in figure and leave through the exit port with the same speed they had when they entered the chamber. The electric field acting on the ions is to have constant magnitude.

Suppose the electric field is produced by two concentric cylindrical electrodes not shown in the diagram, and hence is radial. What magnitude should the field have?
A. `(mv)^2/(2qR)` centered at A
B. `(mv^2)/(2qR)` centered at O
C. `(mv^2)/(qR)` centered at A
D. `(mv)^2/(qR) `centered at O

Answer 1

Correct Answer - C
Concentric cylindrical electrodes will produce radial electric field. As ion is entering at O and leaving at B, hence the path followed by ion should be circular and centrered at A. Required centripetal force should be provided by force on the ion due to electric field. Hence
`qE=(mv^(2))/(R)` or `E=(mv^(2))/(qR)`
As final velocity along x-axis becomes zero and finally the ions starts moving towards y direction the electric field should have component towards x-and y-directions, respectively. for x-component of electric field (using `v_(x)^(2)=u_(x)^(2)+2a_(x)trianglex)`
`0=v^(2)-2((qE_(x))/(m))R` or `E_(x)=(mv^(2))/(2qR)`
for y-component of electric field (again using `v_(y)^(2)=u_(y)^(2)+2atriangley)`
`v^(2)=0+2((qE_(y))/(m))R` or `E_(y)=(mv^(2))/(2qR)` ltbr. Hence net electric field is
`vecE=-(mv^(2))/(2qR)hati+(mv^(2))/(2qR)hatj=(mv^(2))/(2qR)(-hati+hatj)`